Interpolational and extremal properties of L-splinefunctionsCitation for published version (APA):Morsche, ter, H. G. (1982). Interpolational and extremal properties of L-spline functions. Eindhoven: TechnischeHogeschool Eindhoven. https://doi.org/10.6100/IR76507

DOI:10.6100/IR76507

Document status and date:Published: 01/01/1982

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INTERPOLATIONAL AND EXTREMAL PROPERTIES OF Y -SPLINE FUNCTIONS

INTERPOLATIONAL AND EXTREMAL PROPERTIES OF !? -SPLINE FUNCTIONS

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL EINDHOVEN, OP GEZAG VAN DE RECTOR MAGNIFICUS, PROF. IR. J. ERKELENS, VOOR EEN COMMISSIE AANGEWEZEN DOOR HET COLLEGE VAN DEKANEN IN HET OPENBAAR TE VERDEDIGEN OP

VRIJDAG 16 APRIL 1982 TE 16.00 UUR

DOOR

HENRICUS GERHARDUS TER MORSCHE

GEBOREN TE ENTER

Dit proefschrift is goedgekeurd

door de promotoren

Prof.Dr.Ir. F. Schurer

en

Prof.Dr. F.W. Steutel

Aan Marijke,

Judith en Robert-Jan

Met dank aan mijn ouders

CONTENTS

GENERAL I~~RODUCTION

1. BASIC CONCEPTS AND PRELIMINARY RESULTS

1.1. Introduetion and summary

1.2. Notatiens and conventions

1.3. t-spline functions

1.4. Same basic concepts in the theory of t-spline functions

2. THE B-SPLINE FUNCTIONS

2.1. Introduetion and summary

2.2. Fundamentals about B-splines

2.3. The total positivity property of B-splines

2.4. On recurrence relations for B-splines

2.5. The minimum supremum norm of a polynomial B-spline

3. ON RELATIONS BETWEEN FINITE DIFFERENCES AND DERIVATIVES OF

CARDINAL t-SPLINE FUNCTIONS

3.1. Introduetion and summary

3.2. The exponential t-splines

3.3. Relations between finite differences and derivatives

3.4. Some applieations of Theorems 3.3.2 and 3.3.3

4. ON EXISTENCE AND CONVERGENCE PROPERTIES OF I~ERPOLATING CARDINAL

t-SPLINES AND I~ERPOLATING PERIODIC CARDINAL !-SPLINES

4.1. Introduetion and summary

4.2. On the existenee and unicity problem of cardinal t-spline interpolation

4.3. Periadie cardinal t-spline interpolation

4.4. An error estimate for cardinal !-spline interpolation

4.5. An error estimate for periadie cardinal !-spline interpolation

I-IV

5

8

27

27

33

38

44

55

56

67

73

75

76

81

87

93

5. ON THE LANDAU PROBLEM FOR SECOND AND THIRD ORDER DIFFERENTlAL I

OPERATORS

5.1. Introduetion and summary 102

5.2. Optima! differentlation algorithms 105

5.3. Somegeneralproperties of the sets Fm(pn,J} and rm(pn,J,~) 106

5.4. On the relation between the Landau problem and the sets ar (p l and ar+(p J 112 m n m n

5.5. The Landau problem for second order differentlal operators 113

5.6. The Landau problem for third order differential operators 126

6. ON THE LANDAU PROBLEM FOR PERIODIC FUNCTIONS

6,1. Introduetion and summary 132 -6.2. A few preliminary lemmas and the sets F{pn,T) and f(pn,T} 134

6.3. Some properties of the sets F(pn 1 T} and f(pn 1 T) 137

6.4. Perfect Euler l-splines as extrema! functions 144 - 3 6.5. A parametrization of the set êlf(D ,1) 148

7. PERFECT l-SPJ,INES AND THE LANDAU PROBLEM ON THE HALF LINE

7.1. Introduetion and summary

7.2. Perfect l-splines and ~-approximate perfect l-splines

7.3. A representation theerem for the set P (p ) r,.!!_ n

7.4. An extrema! property of perfect l-splines

7.5. A characterization of ar+(p ) m n

REFERENCES

LIST OF SYMBOLS

SUBJECT INDEX

AUTHOR INDEX

SAMENVATTING

CURRICULUM VITAE

151

153

161

167

178

184

192

194

198

200

204

I

GENERAL INTROVUCTION

No doubt, the theory and application of spline functions (splines for

short) has been a flourishing branch of approximation theory during the

last few decades. Although Bernoulli and Euler already used very simple

splines, i.e. polygons, for the approximate salution of differential equa­

tions, it is in general agreed upon that a systematic investigation of

splines began with the work done by I.J. Schoenberg during the secend World

War. It took some time till it was widely recognized that splines have

interesting extremal properties, and are a good tool for the numerical

approximation of functions as well. Since 1960 a large number of papers

have been published. The first bock on the subject by Ahlberg, Nilson and

Walsh [1] dates from 1967 and in recent years a few more have appeared

(cf., for instance, Schumaker [56], which also contains an extensive

bibliography) •

In their original farm splines are piecewise polynomials, in general of

rather low degree, tied tagether with a certain degree of smoothness at the

so-called knots. A natural generalization is obtained if the polynomials

are replaced by functions in the kernel of a linear differential operator

pn(D) of the ferm

where D is the ordinary differentiation operator and the coefficients a 1 (i O,l, •.• ,n-1) are real. The associated splines are then called !­

aplinea; the polynomial splines are obtained if p (D) = Dn. n

There is an extensive literature on the problem of interpolation by means

of polynomial splines. If the knots of the interpolating !-spline and the

points of interpolation, the so-called nodes, are equally spaeed on ~'

then, following Schoenbarg's terrninology, one speaks of cardinal t-apline

interpolation. The interpolation problem may then be formulated as fellows.

Let f be defined on ~, let h be the distance between consecutive knots

(the mesh distance) and let a E (O,h] be prescribed, then one is asked to

determine an t-spline, corresponding to pn(D) and withknotsat O,±h,±2h, ••• ,

II

that interpolates fat the nodes a,a±h,a±2h, •••• Questions concerning

existence and uniqueness of such an .C-spline interpolant, and the problem

of determining (best possible) error estimates naturally present itself.

Cardinal polynomial spline interpolation has been investigated in detail by

Schoenberg [51] in the cases a = h/2 and a = h. Results for arbitrary

values of a € (O,h] in the case of periadie 'cardinal polynomial spline

interpolation are contained in Ter Morsche [40]. Assuming that the charac­

teristic polynomial pn of pn(D) has only real zeros, Michelli [37] has

generalized Schoenherg's results for cardinal polynomial spline interpola­

tion to cardinal .C-spline interpolation.

In the first part of this thesis we study (periodic) cardinal .C-spline

interpolation without the assumption that pn has only real zeros. When

extending the theory to arbitrary operators pn(D) one encounters the

concept of disconjugaay. A differentlal operator pn(D) is said to be dis­

conjugate on an interval (a,b) if pn(D) can be factorized in a sequence of

first order differential operators o 1 ,o2 , ••• ,on of the form Di= winw;1,

where wi is a positive function defined on (a,b)1 apparently, this is true

on any interval (a,b) if pn has only real zeros. In general, it will be

assumed that pn(D) is disconjugate on an appropriate interval, as in the

absence of this proparty some problems are substantially more difficult.

Chapters 3 and 4 deal with existence and uniqueness of an .C-spline inter­

polant, and error estimates are derived that are best possible. The esti­

matas obtained in general are of the form

(x € JR) '

where K :?: 1 is an appropriate constant, 11•11 is the supramum norm on JR, sf

is the .C-spline interpolant to f and f 0 is a so-called perfect Euler .e­epZine, Perfect .C-splines are characterized by the fact that their "pn (D)­

derivative" has constant absolute value c and jumps from ± c to + c at the

knots. Perfect Euler 1:. -splines f 0 addi tionally satisfy the functional

relation f 0 (x+ h) = - f 0 (x) (x e: JR) and are such that their "pn (pl -derivative"

is ± 1. We emphasize that our analysis of the (periodic) cardinal .C-spline

interpolation problem makes essential use of specific relations between

derivatives and finite differences of cardinal 1:.-splines. As a simple

illustration we mention the relation

(i € TL) I

III

which holds for polynomial cubic splines, xi being the equally spaeed knots

and h being the mesh distance. The relations we derive in Chapter 3 are

far-reaching generalizations of (1).

The secend part of this thesis deals with extremal properties of perfect

!-splines in conneetion with so-called Landau probZems. This name has its

crigin in a few interesting inequalities that Landau [30] derived in 1913

for twice differentiable functions. If f and f" are bounded on lR (on

lR; '"' [0,"')) and if we write llfll+ : llflllR+, then 0

llf'll ~ 12 lilrllilf"il , ~ 2lllfll llf''ll , + +

where the constants 12 and 2 are best possible. A generalization of the

first inequality to higher derivatives is due to Kolmogorov [27]; the + corresponding problem on JR0

has been solved by Schoenberg and Cavaretta

[54].

A still further generalization reads as fellows. Let pn be a monic poly­

nomial of degree n, let J c lR be a closed interval, and let m be a positive

number. Further, let Fm(pn,J) denote the set of functions f with f(n-l)

absolutely continuous on every compact subinterval of J, 11 fll J ~ m and

lip (D) fll., ~ 1. The generalized Landau problem then amounts to determining n v .

the best possible upper bound for llpk (D) fiiJ on Fm (pn ,J) , where pk (D) is a

given linear differential operator of order k ~ n-1. The cases J =lR (full­

line case) and J =lR~ (half-line case) are of particular interest. In what

+ fellows we assume that J =lR or J JR0 •

Our analysis of the generalized Landau problem is based on an investigation

of the set

It is shown that rm(pn,J,O} is a compact, convex, subset of lRn having 0

as an interior point. Solving a Landau problem is then equivalent to

maximizing a linear function on rm(p0

,J,O). By the support hyperplane

theerem for convex sets, every boundary point of rm(pn,J,O) has the

property that an appropriate linear function attains its maximum on

rm(pn,J,O) at that point. Consequently, every f E F (p ,J) with the - - ·(n-1) T m n

property that (f(O) ,f' (O), ••. ,f (0)) E 3f (p ,J,O), the boundary of m n .

rm(pn,J,O), is an extremaZ jUnction fora Landau problem, i.e., forsome

pk the function f maximizes llpk (D) f liJ on Fm (p0

,J) • It is therefore of

importance to describe (parametrize) the set arm(pn,J,O). This we do for

IV

general second order and for some specific third order differential opera­

tors in the cases J =~ and J =~~. There is a separate chapter on Landau

problems for periodic functions, where, among ether things, a generalization

of a result of Northcott [46] is given. Landau problems on the half line

are discussed in the last chapter, under the additional hypotheses that pn

has only real zeros and that p (0) = 0. The problem we pose is to minimize n +

liP (D) fll+ with respect to all functions in F (p ,1R0 l satisfying the initial n (i) m n

conditions f (0) =ai (i= O,l, ••• ,n-1) for prescribed ai. It is shown,

by means of a so-called representation theorem, that a perfect l-spline

with a specific oscillation property furnishes the solution.

Landau problems are of importance with respect to the optima! recovery of

the derivatives of smooth functions (cf. Michelli and Rivlin [39,_p. 27]).

The foregoing is intended to introduce the main themes of the thesis and to

give a rough sketch of the problems that are dealt with. We re~rain from

giving here a detailed summary of the contents of the seven chapters.

Instead of this we refer to the first sections of the various chapters

which are introductory and summarize the main results obtained. A few

remarks are in order with respect to Chapters 1 and 2. We have strived to

make this thesis reasonably self-contained. To achieve this, preliminary

material of a general kind is brought together in Chapter 11 most of it is

standard in approximation theory, apart from a generalization of ours of

the Budan-Fourier theerem to piecewise continuous functions. This useful

theorem is, among ether things, applied to give a rather simple proof of

the so-called total positivity property of a sequence of consecutive B­

splines. Chapter 2 introduces the nonpolynomial B-splines and contains

various properties, notably recurrence relations, of these very useful

functions. One of these recurrence relations is used to obtain the knot

distributions for which the supremum norm of a polynomial B-spline is

minimal.

Examples are given in each chapter to illustrate and clarify our assertions.

In order to facilitate the readability of this thesis, at the :end a list of

symbols, an author index and a subject index are added.

1. BASIC CONCEPTS AND PRELIMINARY RESULTS

1. 1. IntJtoduc.tion and J..wnmaJty

The purpose of this chapter is to collect concepts and preliminary results

that will be needed in this thesis. In Sectien 1.2 some general notations

are listed and a number of classes of functions are introduced. Sectien 1.3

gives the definition of an

of some particular

function, tagether with the definitions

such as cardinal !-splines and perfect L-

splines. Sectien 1.4 contains a variety of concepts in the theory of L­

spline functions, e.g. the concept of disconjugacy of a differential

operator, Taylor's formula and Peano's remainder formula, the classical

Budan- Fourier theerem and a generalized version of it, various forms of a

Chebyshev system, the notions of solvent and unisolvent families, and di­

vided differences. Most of them are standard in approximation theory, apart

from the generalized Budan- Fourier theorem. In order to obtain this gener­

alization a specific devi'ce for counting zeros of piecewise continuous func­

tions is used. The generalized Budan-Fourier theerem is applied to !­

splines in Subsectien 1.4.6 in order to catint its zeros.

7.2. NotationJ.. and c.onventionJ..

1.2.1. Formula indication

Formulas are numbered independently from theorems, lemmas '· corollaries and

definitions. When referring, for example, to the third formula of Sectien

2 in Chapter 1, we shall write (1.2.3).

2

1.2.2. Notatiens

Here only those notations are explained that are used throughout this thesis.

A more extensive list of symbols is given on pp. 192,193.

JN

lR

[a,b]

(a,b)

T x

0

V

av

öi,j

f{x+)

f(x-)

[x]

sgn

the set of posi ti ve integers, JN = { 1, 2, 3, .•• } •

thesetof nonnegative integers, JN0 = {0,1,2, ••• }

the set of integers.

the set of real numbers.

the set of complex numbers.

the closed interval {x E lR I a s x s b}

the open interval {x E EI a< x< b}; similarly (a,b], [a,b).

the set [O,oo).

the Euclidean space of dimension n consisting of column vectors.

the transpose of a vector x.

the interior of a set v.

the boundary of a set V.

the empty set,

the Kronecker symbol, i.e., öi . .~

lim f{x +hl. h+O

lim f(x- hl. hi-0

1 and ö. . "' 0 if i >! j. ~.J

the largest integer not exceeding x.

the sign function, i.e.,

sgn(x) [_: (X > 0) 1

(x 0)

(x < 0)

:= is used in a definition if a new symbol occurs on the left-hand

side.

0 marks the end of a proef.

1.2.3. Functions

As we mainly deal with real-valued functions, the range of a function is

assumed to be E, unless otherwise stated.

If f(x) is the value of a function f at x, then the function is denoted by

f or by XI--? f(x) or by f(•).

If f is a function of several variables, say two, then the function of one

variable obtained from f by fixing one of the two variables, say x, is

denoted by YHf(x,y) or by f(x,•).

We praeeed by defining some classes of functions. In these definitions J

stands for an interval and n E ~. unless otherwise stated.

C(J)

c(n) (J)

PC(J)

AC(n) (J)

the set of continuous functions defined on J.

the set of functions f defined on J having continuous n-th

derivatives, i.e., f(n) c C(J) where n c ~0 •

the set of functions f defined on J that are continuous on J

with the possible exception of finitely many points in any

bounded subinterval of J, and such that at every point x of

discontinuity f(x+) and f(x-) both exist with f(x) f(x+).

the set of functions f E c<n-l) (J) for which the (n-1)-st

derivatives f(n- 1) are absolutely continuous on every compact

subinterval of J.

3

PC(n) (J) thesetof functions f E AC(n) (J) for which the (n-1)-st deriva-

L"'(J)

w<nl (J)

t . f(n-l) · t 1 f f · · ( ) · f Lves are Lil egra s o unctLons Lil PC J, L.e., or

f E Pc (n) (J) there · t f t' PC(J) h th t every eXLS s a unc LOD g E suc a

t

f(n-1) (t) f<n- 1) tt0

J + J g(T)dT ( t E J t t0

E J) •

to

The n-th derivative is defined on J with the possible exception

of finitely many points in any bounded subinterval of J.

At any point t E J we define f(n) (t) : g(t).

the set of measurable functions f which are essentially bounded

on J, i.e., for every functlon f E L."(J) there exists a number

M > 0 such that I f(t) I < M (a.e.) .

thesetof functions f E AC(n) (J) for which f(n) E L",(J).

4

1.2.4. The supremum norm of a function

If a function f is essentially

I fl on J is denoted by .llf!IJ. + If J = JR or J = JR0 then the

bounded on J, then the essential supremum of

notatien for the supremum norm is shortened

by writing llfll := llfiiJR, llfll+ :=

1.2.5. Interpolation of Hermite data

Let!= {t0,t1, ••• ,tm-- 1lT € JRm with t 0 ~ t 1 ~ ••• ~ tm-l' If fis a func­

tion that is sufficiently often differentiable, then f{t) = f(t0 ,t1, ••• ,t 1> T -- - m-

denotes a vector.;:.= (a0 ,et1, ••• ,etm_ 1l € JRm defined as fellows.: if

t. 1 < t. t. 1 ••• = t.+k 1 < tj k' then et .. := f(i) (t.) J- J J+ J - + J+~ J

(i O,l, ••• ,k-1), in particular, if the tj are distinct then aj := f(tj)

(j 0,1, ••• ,n-1). The sequence ,tj+l'''''tj+k-1 with

tj-1 < tj = tj+1 = of length k.

tj+k- 1 < tj+k will be called a aoinaident bloak

We say that a function f interpolates the data (y0 ,y1, ••• ,ym-l) at the

points t 0 ,t1, ••• ,tm-l when !<t0,t1, ••• ,tm_1l = (y0 ,y1, ••• ,ym-l)r. This

kind of interpolation is called He~ite interpolation.

1.2.6. Determinants

T m Let!= (t0 ,t1, ... ,tm-l) EJR with t 0 s t 1 s ... s tm-land let

~0 ,~1 , ••• ,~m- 1 be m functions that are sufficiently aften differentiable,

then

: = det <.Po I • • • I .Pm-- 1) ,

where det {.Po I • • • I .Pm_1) is the determinant of the m x m matrix, the j-th

column of which is given by ~j(t0 ,t1 , ••• ,tm_ 1>.

Usually, spline functions (throughout the thesis the term "spline" will be

used as a synonym for "spline function") are defined as functions with a

certain degree of smoothness consisting of piecewise polynomials tied

tagether at the so-called knots. More generally, the polynomials are re­

placed by functions satisfying a given linear homogeneaus differentlal

equation! f 0 (cf. Michelli [37]) • This leads to the de fini tion of an

! -spline.

The space of all polynomials with real coefficients and of degree at most

n E ~ is denoted by "n· A pelynemial pn in "n is called manie if pn has

degree n and if its leading coefficient is equal te ene.

Let pn E: Tin be monic, i.e., pn {z) z n n-1

then the + a z + ... + ao,

linear differentlal operator f 1-4 f (n) n-1 (n-1)

+ is denoted + an-lf + ... by Dn + a Dn-1 + ... + a0r or by p (D) for short, with Df := f' and

Dkf n-1 n

Df(k-1) (k ;:> 2).

The kernel of the operator pn(D), denoted by Ker(pn) or Ker(pn(D)), is

defined as fellows.

DEFINITION 1.3.1.

:= {f E AC(n) (lR) I p (D)f(t) n

0 (a.e.)} •

DEFINITION 1.3.2. Lets bedefinedon [a,b]. We say that sis an l-spZine

funation of order n, if there exists a manie poZynomiaZ pn E "n and a

finite sequenee of points x 1,x2, ••• ,~ with a= x0 < x 1 $ x 2 •.. $ xk <

< ~+1 band x1 < xi+n (i= l, ••• ,k-n), sueh that

i) on every nonempty interval Cx1 ,xi+l) (i

coincides with an element of Ker(p0),

O,l, ... ,k) the function s

5

ii) if 1 < x1 = xi+l = . . . 1 < xi+j' i.e., if x1 has muZtipZicity

j, then s has a continuous (n-1-j)-th derivative in a neighbourhood of

j

If pn (z) zn then the !-spline s consists of piecewise polynomials and it

will be called a poZynomiaZ spline of order n (or of degree n-1).

The points x 1,x2 , ••. ,~ are called the knots of the !~spline.

6

If a knot has multiplicity one, then it is called a simple knot. Occasional­

ly, the distinct points among the knots x1,x2, .•• ,~ are denoted by

y 1,y2, ••• ,yk and their multiplicities by v1,v2, ••• ,vk, respectively.

We also deal with .C-splines defined on E or on E;. In these cases the

number of knots is allowed to be infinite;, however, a finite accumulation

of knots is not permitted.

The following definition concerns an !-spline function that is fundamental

in the theory of splines.

DEFINITION 1.3.3. Let~ E Ker(pn) be the jUnation satisfYing

, (n-2) ~(0) = ~ (0) = ••• = ~ (0) = 0 1

~ (n-1) (O) 1 •

Then the funation ~+ E PC(n-l) (lR) is defined by

(

(!1 (t)

~+(t) := 0

(t ~ 0) ,

(t < 0)

We call q~ the fundamentaL funation corresponding to the operator pn (D) and

(!1+ the Green's funation corresponding to pn(D). The dependenee of the func­

tions ~ and ~+ on pn is not expressed in the notation.

The function ~+ can be used to represent .C-splines. For instance, if s is

an .C-spline with knots located at y 1,y2 , ••• ,yk with y 1 < y 2 < ••• < yk and

with multiplicities v1,v2, ••• ,vk, respectively, then scan be written as

(1.3.1) k V·-1

s ct> = g ct> + I ~ w. j < j > ct - y i> , 1=1 j=O J.,

where g E Ker(p ) and the real numbers w. j are uniquely determined (cf. n J.,

Karlin [21, p. 517]).

For numerical purposes the representation by means of ~+ can be bad (cf.

De Boor [5, p. 104]). The so-called B-spline functions constitute a basis

for .C-splines that is appropriate for both practical and theoretica! pur:­

poses. Chapter 2 is devoted to these functions.

We shall give special attention to .C-splines with equally spaeed simple

knots. With respect to this class the following definition is in order.

7

DEFINITION 1.3.4. Anf.-spline function sconesponding to the opemtor pn(D)

is called a cardinal .C -spline function of order n E .N if there exists a

number h > 0 such that

i) pn(D)s(t) =0 (ih< t< (i+1)h; iE:iZ),

ii) sE c<n-2

) (JR) if n;:: 2 and sE PC(JR) if n 1.

The number h in the de fini ti on of a cardinal .,C -spline is called the mesh

distance of the knot distribution.

Thesetof all cardinal .C-splines corresponding to the operator pn(D) and

with mesh distance h is, in genera!, denoted by

( 1. 3. 2)

occasionally, the notatien S(pn(D) ,hl will also be used.

Usually, the term cardinal is reserved for l-splines with knots at the

integers, but by a simple transformation of scale t = hT the knots are

transformed to the integers, while the differential operator pn(D) is

replaced by pn(hD).

The cardinal pclynomial splines are studied in detail by Schoenberg [51] in

his monograph. Parts of the theory are extended by the author of this thesis

in[40] and, independently, by Michelli [37].

Finally, the so-called perfect l-splines will be defined.Thesesplines often

occur as "extremal functions" with respect to certain extremal problems in

the space W(n) (J) in case the supremum norm is used (cf. Chapters 5, 6, 7).

DEFINITION 1.3.5. A function s, defined on the interval J, is called a

perfect l-spline fUnction corresponding to the operator pn(D) if sis an

l-spline of order n + 1 corresponding to the operator Dpn (D) with the

property that a number c exists such that for all t E J, knots being ex­

cluded,

IP<D)s(tll c. n

It follows from this definition that the "pn (D) -derivative" of s has con­

stant absolute value c and jumps from ± c to + c at the knots. Note that a

perfect .C-spline correspcnding to the operator pn (D) i·s an l-spline of

order n + 1 corresponding to the operator Dpn {D) •

8

1. 4. Same bM-<..c. c.oi'U!ep.U bt .the .theoi!.!J oó .c -.&p.Une 6unc..:üo~

In order to make this thesis reasonably self-contained, in this section we

will collect some results that will be frequently used. In general, proofs

are omitted in case a reference is available or if it concerns! proofs that

are standard in the theory of .C-spline functions.

1.4.1. Disconjugacy of a differential operator

DEFINITION 1.4.1. The differentiaZ operator pn{D) is said to be disaonjugate

on an intervaZ J if there exist n striatZy positive jUnations wiE c<n+1-i) (J) (i 1,2, ••• ,n) suah that pn(D) aan be faatorized as

(1.4.1)

where the differentiaZ operator Di is defined by

(1.4.2) {i 1, 2, ... ,n)

The interval J is called an interval of disconjugacy.

The following characterization was proved by Pólya in 1922.

LEMMA 1.4.2 (Pólya[48]). The differentiaZ operator pn(D) is disconjugate on

an open intervaZ {a,b) if and onZy if no nontriviaZ funetion f E Ker(pn)

has more than n- 1 zeros in (a,b), aounting muZtipZiaities.

If pn has only real zeros, i.e., if pn (D) == (D- À1I) ••• (D- Àni} with

Ài E ~. then one has

so p (D) is disconjugate on the whole real line. If p has at ;least one n n nonreal zero, then an interval of disconjugacy is finite. As an example

let us consider the operator p 2 (D) = D 2 +I, One easily verifies that for

all a E :Rand f E AC( 2) (lR)

1 2 f(t) p2(D)f(t) = sin(t-a) 0 sin (t-a) 0 sin(t-a}

Hence intervals of the form (a,a+n) are intervals of disconjugacy for

p2 (D) •

Since pn(D) has constant real coefficients, the maximal length ~ of an

interval of disconjugacy does not depend on the location of that interval

on the real line. It turns out (cf. Troch [62, Theorem 1]) that the maximum

intervals on which each nontrivial function in Ker(pn) has at most n- 1

zeros counting multiplicities, are half open intervals [a,a+~) or (a,a+t].

For second order differential operators the maximal length ~ is simply the

distance between two consecutive zeros of the corresponding fundamental

function (!). For third order differential operators the function (j) also

determines the number ~. This is a consequence of Lemma 2 in Troch [62] 1

which for third order differential operatorscan be statedas follows.

9

LEMMA 1.4.3 (Troch[62]). Let p 3 E n3 be a monic poZynomial having a nonreal

zero, and let x0

E R be a nearest point to zero for which (!) (x0

)

x0 ~ o. Then ~ = lx0 1. 0 and

If the order of the differential operator exceeds three 1 the zeros of its

fundamental function !Jl do not in general determine the number t as in the

case of second and third order differential operators. This is shown by the

following example.

4 EXAMPLE. If p4 (D) = D -I, then !Jl (t) = ~ (sinh t - sin t) and thus (j) (t) ~ 0

when t f. 0. As t~sin tE Ker(p4), by Lemma 1.4.2 the number ~ for p 4 (D)

is finite.

It is well known that a trigonometrie polynomial of degree n, i.e., a func­

tion belonging to Ker(p2n+l} with

( 1. 4.3) P2n+l (D) 2 2 2 2 D(D +I) (D +4I) ••• (D +n I) 1

cannot have more than 2n zeros in an interval [a,b) with b-a~· 2n 1 unless

it is identically zero. As shown by the function t~ sin(nt) 1 the number t

for the operator (1.4.3) is equal to 2n.

For more detailed information concerning the property of disconjugacy the

reader is referred to the hook of Coppel [14].

10

1.4.2. Taylor's formula and Peano's remainder formula for pn(D)

For the differential operator pn(D) Taylor's formula can be statedas

fellows.

LEMMA 1.4.4. Let f E AC(n) ([a,b]) and tet.g E Ker{pn) be the unique funation • .., • (i) { ) (i) { ) (. 0 1 1) Th sat~sJy~ng g a = f a ~ = 1 1••·,n- • en

'{1.4.4) f(t) g(t) + r qJ+(t-T)pn(D)f(T)dT

a

(a ~ t ~ b) ,

UJhere the funation rp+ is given by Defin:ition 1.3.3.

We preeeed by giving Peano's remainder formula for pn(D).

LEMMA 1.4.5. Let À be a linea!' funationa'l ckfined on AC(n} ([a,b]) of the

fo!'m

À(f) (f E AC(n) ([a,b])) 1

UJhere m E :N• n € :NJ aR.1

j E JR. and a s x1 < x2 < • • • < x s b. m If À(g) 0 for all g E Ker(p >. then one has n

b

{1.4.5) À {f) I K(T)pn(D)f(T)dT 1

a

UJhere the kemel K is ckfined by K(T) := À (t- <p + (t- T)).

A proef of Lemma 1.4.5 if pn(D) on is contained in Davis [ 161 p. 70 J •

In the case of a general differential operator pn(D) a proof of Peano's

remainder formula can be given along similar lines using Taylor's formula

of Lemma 1.4.4. We note that Lemma 1.4.5 does not state Peano's remainder

formula in its full generality.

1.4.3. The theerem of Budan-Fourier

The classical form of the Budan-Fourier theerem gives an upper bound for

the number of zeros in an open interval (a,b) of a polynomial of degree n {i) n

by means of the number of sign changes in the sequences (p (a))i=O and

(p (i) (b)) ~:O (cf. Karlin [ 21, p. 317]) • It reads as follows:

THEOREM 1.4.6 (Budan-Fourier). Let p E 'lfn be a poZynomiaZ of exaat

n EN. The total number of zeros of pin (a,b), aounting muZtipZiaities,

is bounded by

(n) + (n) (p(a) , ... ,p (a)) - S (p(b) , ... ,p (b)) ,

where and s+ are defined by Definition 1.4. 7.

11

An extension to polynomial splines with simple knots is given by De Boor

and Schoenberg [6, p. 6] and to polynornial splines with knots of a~bitrary

rnultiplicity by Melkman [36].

The version of the Budan-Fourier theerem we present here is a generaliza­

tion of the classical theerem to the set of functions PC(n) ([a,b]). Before

we can forrnulate our result we have to agree on the way of counting zeros.

This gives rise to the following definitions.

DEFINITION 1.4.7. Let a1,a2 , ••• ,an be a sequenae of reaZ numbers. Then

denotes the number of sign changes in a 1 ,a2 , ••• ,an by deZeting zero

entries. The maximum number of sign changes that aan be obtained by re­

pZacing the zero entries in a 1 ,a2 , ••. , by +1 ar -1 is denoted by

DEFINITION 1.4.8. Let f be a function defined on (a,b). Then

We shall call S-{a1,a2, ••• ,an) the nurnber of strong sign changes and

S+(a1,a2, ••• ,an) the nurnber of weak sign changes of a 1 ,a2, .•. ,an.

12

One has the following equality (cf. De Boor and Schoenberg [6, p. 3]):

(1.4.6)

The next step is to define the way of counting zeros in an open interval

(a,b) of functions in PC(n) ([a,b]) (n = 0,1, ••• ). This will be done as

follows. First the procedure is described for the class PC([a,b]); one then

extends it to the class PC(n) ([a,b]).

As zeros in (a,b) of a function f € PC([a,b]) are considered: points

x0 € (a,b) where f(x0-)f{x0 l s 0 and intervals [c,d] c {a,b) such that fis

identically zero in (c,d).

The function f is said to "vanish" at a point x0 if x0 is a zero of f as

defined above; so f may vanish at x0 even when f(x0 ) ~ 0. The function f is

said to "vanish identically" in an interval J if f vanishes at every point

of J.

We shall only take into account those zeros a (a point x0 or an interval

[c,d] c (a,b)) having the property that f does not vanish identically in

any interval containing {x0 } {or [c,d]) as a proper subset. These zeros

will be called strong zeros.

DEFINITION 1.4.9. Let f € PC([a,b]) and Zet a he astrong zero of fin

{a,b). Then the multipliaity M0

(f,a} of a is determined as follows.

If S-{f,U) = oo in eaah neighbourhood u c (a,b) of a, then M0

(f,a) := ~.

If s- (f,U} = 1 for aU sufficiently small neighbourhoods u c (a,bl of a~ then M0 (f,a) := 1.

If s- {f,U) = 0 for all suffiaiently smaU neighbourhoods u c (a,bl of a, then M0 (f,a) := 0.

Note that in this definition oo, 1 and 0 are the only possible values of

S-(f,U). Clearly, fora sufficiently small neighbourhood U of a streng

zero a we have either s- (f,U) = 0 or s- (f,U) = 1 or s- (f,U) = "' • According

to the counting rule as given above and taking into account Definition

1.4.8, the total number of strengzerosin {a,b), counting multiplicities,

of a function f € PC([a,b]) is equal to S-(f,(a,b)).

The following example illustrates the way of counting zeros for functions

f € PC{[a,b]).

13

~"--------- . L_~ 0 1 2--~ 5 6....---7 !.,_../9

Figure 1.

The streng zeros are 2, 3, [4,5], 6, [7,8] with multiplicities 0, 1, 1, 1, 0,

respectively. Hence s (f,(0,9)) ~ 3.

Our next purpose is to define the multiplicity of a strong zero of a func­

tion f E PC(n) ([a,b]) (n ~ 1) with respect to a sequence of differential

operators o1,n2, ••• ,Dn given by

(1.4. 7) 1

D. ~ w. D-l. l. w.

l.

(i 1, ..• , n) ,

where w. E C(n+1-i) ([a,b]) is a strictly positive function. l.

[iJ We define the operators D as fellows:

(1.4.8) D[O] := I , D[i] :~Di D ••• D i-1 1 (i 1,2, ... , n) •

(n) ] DEFINITION 1.4.10. Let f E PC ([a,b l (n ~ 1} and let a be astrong zero

of fin (a,b). The multipliaity of a with respect to the sequence of dif-n ferentiaZ operators n 1,n2 , •.• ,Dn, denoted by M((Dil 1,f,a), is defined as

foZlCJWs. If D[i]f(a) 0 for i

n M( (Di) 1,f,a) := k

o[i]f(a) = 0 for i n M((Di) 1,f,a) := n.

0,1, ••• ,k-1 and D[k]f(a) # 0 with k ~ n-1, then

[n] O,l, ••• ,n-1 and D f does not vanish at a, then

If D[i]f vanishes at a for i

REMARK 1. If a is astrong zero of f E PC(n) ([a,b]) with an odd multiplicity

M((D1l7,f,a) then f changes sign at a, and if M((Di)~,f,a) is even then f

does not change sign at a.

REMARK 2. If a is a streng zero of f E PC(n) ([a,b]) with M((D1 )~,f,a) =

= k ~ n-1, then, of course, a reduces to a single point x0 and 1 (k-1} (k)

f(x0 J = f (x0) = ••• = f (x0) 0, f (x0) # 0. Moreover, x0 then is

an isolated zero, i.e., there exists a neighbourhood of x0 such that f

14

vanishes in U only at the point x0 •

on the other hand, if x0

e (a,b) satisfies f(x0) = f'(x0J = .•. = f(k- 1) (x

0J = 0 and f(k) (x

0) f 0 with k ~ n-1, then it is easy to verify

n . that M((Di) 1,f,a) = k for any sequence of differentlal operators

D1,D2, ••• ,D0

given by (1.4.7).

REMARK 3. We show that there may exist two different sequences of differ-~ n n

ential operators (Di) 1 and (Di) 1 such that for some function

f e PC(n) ([a,b]) and some x0 e (a,b) M((Di)~ 1 f,x0 J E {n,~+1}, whereas n

M((Di) 1,f,x0 ) = ""• To this end we take n = 1, D1 = D and D1 = w1 Dw1

with

w1 given by

J 1 + ( I x I ~ 1, x f 0) I

w 1 (x) := l1 (x = 0) •

If f E PC(l) ([-1,1]) satisfies f(O) = 0, then

and thus

f<x> = w1 <x> r w;1

<t>n1f<tldt

0

- Jx -1 -f'(x) = D1f(x} + wÎ(x) w1 (t)D1f(T)dT

0

2 Substituting a function f for which D1f(x) =x cos (1/x), we may easily

verify that M((D1J,f,O) = 2 and M((D) ,f,O) = ~.

REMARK 4. If a is a strong zero of f € PC(n) ([a,b]) (n ~ 1) with n { } - [n] - [n] M((D1J 1,f,a) E n,n+l, then S (D f,U) = 0 or S (D f,U) = 1 in a neigh-

bourhood U of a. From this it follows that a is an isolated zero.

The total number of strong zerosin (a,b) of f E PC(n) ([a,b]) (n E ~0), counting multiplicities according to Definitions 1.4.9 and 1.4,10, is

denoted by

(1.4.9)

Note that if n = 0 then (1.4.9) may be replaced by S-(f,(a,b)).

15

Let f E PC(n) ([a,b]). Taking into account (1.4.8) one has [ '] (n- . )

D J f E PC J ([a,b]), and the total number of streng zeros in (a,b) of

D[j]f, counting multiplicities in accordance with Definitions 1.4.9 and

1.4.10, is denoted by

(1.4.10) (j 0, 1 , ••. , n) •

Note that if j - [nJ

n then ( 1. 4 .10) may be rep la eed by S (D f, (a ,b)) .

The following lemma may be considered as a generalization of Rolle's

theorem.

LEMMA 1.4.11. Let f E PC(n) ([a,b]) (n ;" 1). Then

(1.4.11) (n EN) •

PROOF. Let a 1 and a 2 be two distinct streng zeros of f. Assume that a 1 and

a 2 are points x 1 and x 2 with x1 < x 2• Since

0

x2

f(x2

l = w1

(x2J J ---1--- D[l]f(T}dT

W l (T) 0 ,

and f does not vanish identically in (x 1 ,x2), it follows that - [1] s (D f,(x1 ,x2JJ ;" 1 and therefore

If a 1 is a streng zero of multiplicity k ;" 2 of f, then it fellows immedi­

ately from Definition 1.4.10 that a1

is a zero of multiplicity k- 1 of

D[l]f. This proves our lemma, since the case that at least one of the

strong zeros is an interval can be treated in a similar way.

COROLLARY 1.4.12. Let f E PC(n) ([a,b]) (n ~ 1), then

(1.4.12) n - [n]

Z((D1

J 1,f, (a,bll ss (D f, (a,b)) +n

PROOF. obviously, the inequality

(1.4.13) n [j]

Z((D.). 1,0 f,(a,b)) ~ J+

0

16

is a direct consequence of Lemma 1,4.11. Repeated application of (1.4.13)

yields (1.4.12).

In the proef of our version of the Budan-Fourier theerem the following

lemma is needed.

LEMMA 1.4,13. Let f E PC(n) ([a,b]), If D[n]f(a) ~ 0, then

(1.4.14) - [1] [n]

S (f(a),D f(a), ... ,D f(a)) =

+ [1] [n] lim S (f(x),D f{x), ••• ,D f{x)) x.j.a

If o[n]f{b-) ~· 0 then

(1.4.15} + [1] [n]

S (f(b) ,D f(b), ... ,D f(b-})

+ [ 1] [n] = lim S {f(x),D f(x), ••• ,D f(x-)) xtb

PROOF. Since D[n]f(a) ~ 0 there exists a number E1

> 0 such that [i] - [n]

0

D f(x) ~ 0 (i= O,l, ••• ,n; a< x< a+E 1). Hences (f(x), ••• ,o f(x))

= S+(f(x), •.• ,D[n]f(x)). If Ó[j+l]f(a} = D[j+2]f(a) = ••• = D[j+r-l]f(a) =0, while D[j]f(a)D[j+r]f(a) I O, then sgn(D[j+k]f(x)) = sgn{D[j+r]f(a))

[ '] [ ·] (k = l, ••• ,r) and sgn(D J f(x)) = sgn(D J f(a)), As a consequence we have

for x E (a,a+E1

)

(1.4.16) - [j] [j+r] + [j] [j+r] s (D f (a}, ••• ,D f (a)) = S (D f (x) , .•• ,D f (x)) .

[n] Application of (1.4.16) to consecutive segmentsof (f(a), ••. ,o f(a))

yields (1.4.14),

If D[n]f(b-) ~ 0 then there exists a number E2

> 0 such that D[i]f(x) ~ 0 . [j+l]

(L = 0,1, ••• ,n; b-E2 <x< b). Furthermore, if D f(b) = ••• =

D[j+r-l]f(b) = 0 and D[j]f(b)D[j+r]f(b) I 0, then sgn(D[j+k]f(x))

= sgn((-1)r+k D[j+r]f(b-)) and sgn(D[j]f(x)) = sgn(D[j]f(b))

(k = 1,2, ••• ,r; b-E2

<x< b). Hence

+ [j] [j+r] S (D f(x), ••• ,D f(x)) ,

which implies (1.4.15). 0

17

We remark that a particular case of the previous lemma, i.e., when

D[i] =Di (i= 1,2, ••• ,n), can be found in De Boor and Schoenberg [6,p. 6],

in Karlin and Michelli [24], and in Schumaker [56, p. 163].

THEOREM 1.4.14 (Generalized Budan-Fourier theorem). Let f E PC(n) ([a,b])

and Zet Di and D[i] he given by (1.4.7) and (1.4.8), respectively.

If D[n]f(a)D[n]f(b-) I O, then

( 1.4 .17)

- [n] - [1] [n] s: S (D f, (a,b)) + S (f(a) ,D f(a), ... ,D f(a)) +

+ [1] [n] -S (f(b),D f(b), ••. ,D f(b-))

PROOF. First we prove (1.4.17) under the additional condition that

(i= 0,1, ••• ,n-1)

We assert that for j = O,l, ••• ,n

(1.4.19) n [ j]

Z((D.). 1

,D f,(a,b)) S: ~ J+

n [j+l] ::;; Z({Di)j+

2,D f,(a,b)) +

Varying n in (1.4.19) one can see that it suffices to prove (1.4.19) for n [1] n

j=O. If Z{(Di) 2 ,D f,(a,b)) ~ 1 and Z((Di) 1 ,f,(a,b)) 0 then (1.4.19)

holds trivially. Now let us assume that

n Z( {Di) l ,f, (a,b)) 0 .

Then sgn ( f (a)) [1] [1]

sgn(f(b)) and sgn(D f(a)) = sgn(D f(b-)) and again

(1.4.19) holds trivially. So, the case Z((Di)~,f,(a,b)) ~ 1 remains.

Since f is continuous we can find points c and d in (a,b) such that n [1]

f(a)/f(c} > 1 and f(b}/f(d} > 1. If Z((Di} 2 ,D f,(a,c)) 0, then

f(c) > f(a) in case D[ 1]f(a) > 0 and f(c) < f(a) in case D[l]f(a) < 0,

Hence S+(f(a) ,D[ 1]f(a}) = 1. A similar reasoning applied ~o the interval n [ 1]

if Z( (Di) 2,D , f, (d,b)) = 0, then (d,b) leads to the following assertion:

S+(f(b),D[ (b-}} = 0. So we have

18

and

These inequalities combined with (1.4.11) give

We thus obtain (1.4.19). Repeated application of (1.4.19) yields

n - [ n] n-l [ ] [ 1] Z((Di)

1,f,(a,b)) s; S (D f,(a,b)) + I S-(D j f(a),D j+ f(a)) +

j=O

In view of this and taking into account condition (1.4.18), inequality

(1.4.17) follows from the identity

If condition (1.4.18) is not satisfied, then because of the assumption

D[n]f(a)D[n]f{b-) # 0 it follows that for all sufficiently small e: > 0 one

bas D[l]f(a+E)D[i]f(b-e:-) # 0 (i= O,l, ••• ,n). Therefore, (1.4.17) holds on

the interval (a+e:,b-e:). Since Z((Di)~ 1 f,(a,b)) = Z((Di)~ 1 f,(a+e:,b-e:)) for

sufficiently small e: > 0, Theorem 1.4.14 is obtained by letting e: tend to

zero and using Lemma 1.4.13.

REMARK. Theorem 1.4.14 implies the classica! Budan-Fourier theorem 1.4.6;

this can beseen as fellows. Since p(n) (a) I 0 one has that

S-(p(n) ,(a,b)) 0. Furthermore, the multiplicity of a zero of p with n respect to the operators (Di) 1 is as usual determined by consecutive

derivatives. Hence {1.4.17) yields Theorem 1.4.6,

0

1.4.4. Chebyshev systems

DEFINITION 1.4.15. Let J be an intePval and let ~0 ,~ 1 , ••• ,~n-l ben func­

tions in C(J), The set {~0 ,~ 1 , ••• ,~n-l} is called a Chebyshev system on J

if (cf. Beetion 1.2.6 for notation)

> 0

tor each choice of tr~ n pointstiE J with t0

< t1

< ••• < tn_ 1 •

If the determinant in the foregoing definition is nonnegative for all

t 0 < t 1 < ..• < tn_ 1 and if the functions ~ 0 ,~ 1 , ••. ,~n- 1 are linearly in­

dependent, then {~ 0 ,~ 1 , ••• ,~n- 1 } is called a weak Chebyshev system on J.

19

DEFINITION 1.4.16. The Chebyshev system ~0 ,~ 1 , •.• ,~n- 1 is called an ex­

tended Chebyshev system of orderpon J if ~- "c(p-l) (J) (i 0,1, ••• ,n-1) ~

and if

> 0

for all t 0 :s: t 1 s ••• s tn_ 1 (tiE J)~ with coincident bloeks of Length

not exceeding p (cf. Section 1.2.5).

If the determinant in Definition 1.4.16 is nonnegative for every choice of

:S: t 1 :S: ••• s tn_ 1 with coincident blocks of length not exceeding pand

if the functions ~0 ,~ 1 , ••• ,~n- 1 are linearly independent, then these func­

tions are called an extendedweak Chebyshev system of orderpon J.

In what follows we give examples of Chebyshev systems together with a

number of lemmas that will be used in the sequel of this thesis.

EXAMPLE 1. If ~ is the fundamental function corresponding to the operator

pn(D), then the functions ~~~·, ••• ,~(n-l) are linearly independent on every

interval J, but they only form an extended Chebyshev system of order n on

intervals where p (D} is disconjugate. This assertien follows from Theorem n ,

4.3 in Karlin and Studden [25, p. 24]. An important consequence of the (n-1)

fact that ~.~·, ••• ,~ form an extended Chebyshev system on intervals

of disconjugacy is given in the following lemma.

20

LEMMA 1.4.17. Let (a,bl be an interval of disaonjugaay fo~ the ope~ato~

pn(Dl, let a< x 1 ~ ••• ~ xn <band let a 1,a2 , ••• ,an be ~eal numbe~s. Then

the~ ereiste a unique jUnation f é Ker(pnl auah that

. T !(xl ,x2, ••• ,xnl = (al ,a2, •. • ,an) ,

i.e.~ f ia the Hermite interpolate of the data a1 ,a2 , ••• ,an (af. Seation

1.2.5).

PROOF. Since

(n-1) ql

x n

> 0 ,

the null function is the unique function interpolating zero data. This

guarantees the existence and the uniqueness of a function f E Ker(pn) with

the required properties. 0

LEMMA 1.4.18. Let (a,b) be an inte~val of disaonjugaay fo~ the ope~to~

pn(D) and let n distinat points x1 < x2 < ••• < xn be given in (a,b). Let

q1 be the fundamental jUnation of pn (Dl and de fine the funations 'Pi

(i= 1, ••• ,n) by

(1.4.20) epi (t) := cp(t-x1

l (t € lR) •

Then the jUnations ~P 1 , ••• ,cpn form an extended Chebyshev syetem of o~de~ n

on every intewal of disconjugaay fo~ the ope~ator pn (D).

PROOF. Only a sketch of the proof will be given. First we show that the

functions cp 1 , ••• ,qln forma basis forthespace Ker(pn). If not, then there

would exist constants a 1 , •.• ,an with (a1 , ••• ,an) ~ (0,0, ••• ,0) such that

and we would have

cp

0 x -x n n-1

(n-1) ql

x -x n 1

(j = O,l, ••• ,n-1) ,

0 ,

which contradiets the fact that ~~~·, ••• ,~(n- 1 ) is an extended Chebyshev

system on the interval (O,b-a) (cf. Example 1, p. 19).

21

By a nonsingular linear transformation the basis ~~~·, ••• ,~(n- 1 ) can be

transformed into the basis ~ 1 , •.• ,~n· It then suffices to note that non­

singular linear transformations transfarm Chebyshev systems into Chebyshev

systems.

A variant of the above lemma is obtained by dropping the assumption that

x 1 , ..• ,xn are distinct. If there is a coincident bleek xi-l < xi

= xi+j-l < xi+j' then the functions ~i'~i+ 1 , ..• ,~i+j- 1 in (1.4.20} only

have to be replaced by

(1.4.21) (k = 0 t 1 t • • • 1 j-1 1 t E :R) •

Then ~ 1 , ••• ,~n again form an extended Chebyshev system of ordernon every

interval of disconjugacy.

EXAMPLE 2. Let m distinct points x1 < x 2 < ••. < xm be given and let

[x1,xm] be an interval of disconjugacy for the operator pn(D}. The func­

tions ~i (i = 1, ... ,m) are defined by ~i (t} := ~+(t- xi) (t E :R). Then on

every interval of disconjugacy the functions ~ 1 , ..• ,~m farm an extended

weak Chebyshev system of order n-1.

This assertien is a consequence of a result of Karlin, which for this par­

ticular case takes on the following form.

IJ

LEMMA 1.4.19 (Karlin [21, p. 503]). Let (a,b) be an interval of discon.jugacy

for the operator pn(D). Let x0 5 x1 5 ••• 5 xm be points located in (a,b)

with x. <x.+ (i= O,l, ••• ,m-nJ. Furthermore, let t 0 ,t1, ••• ,t satisfy the ~ ~ n m

conditions

ii) < ti+n (i= 0,1, ••• ,m-n) ,

iii) if ti-l< ti= ti+l = ••• ti+j-1 < ti+j and ~- 1 < ~ = xk+1 =

~H-l < xkH for i,j ,k,R. in :N and if ~ ti• then R.+j 5 n+l.

Finally. let the functions ~0 , ••• ,~m be defined by ~i+R.(~)

if xi-1 < xi = xi+l = ••. = xi+R.'

Th en

22

(1.4.22). ::1 ~ 0 •

Moreover, for n ~ 2 strict inequaUty in (1.4.22) holde if and only if

where xj :="' if j > m.

For n = 1 strict inequality holds if and only if xi :S. ti < xi+l

(i= O,l, ••• ,m),

If the computation of (1.4.22) requires the (n-1}-st or n-th derivative of

q1+ at zero we define fP!n-l) (0) := fP (n-1) (0) and IP!nl (0) := rp (n) (0).

1.4.5. Solventand unisolvent families of functions

DEFINITION 1.4.20. Let n c En, let J be an interval and let F0 (J) be a set

of functions f(~;·l indexed by ~ E o. The set F0 (J) is called a solvent

family of order p if

i) f (~;.) € c (p-l) (J) (~ € Q) ,

iil for any sequence of n data a. 1,a. 2 , ••• ,a.n and any sequence of n points

x0 :S. x1 :S. ••• :S. xn- 1 with coincident bloaks of Zength not exaeeding p

there exists a point~ E 0 such that !!~;x 1 ,x2 , ••• ,xn) = (a. 1 ,a2 , ••• ,an)T.

If ~ E Q in ii) is uniqualy determined, then F0 (J) is said to be a uni­

solvent family of order p.

The above definition is contained in Pinkus [47, p. 89]. An e~ample of a

unisolvent family of order p is the linear span of an extended Chebyshev

system of order p; an example of a solvent family consisting of a partic­

ular set of t-splines shall be given in Chapter 7, p. 154.

23

1.4.6. The number of zerosof an !-spline function

Rolle's theerem is one of the basic tools in the theory of polynomial

interpolation. When counting zeros of !-splines we have to deal with situa­

tions where the given !-spline may be identically zero on a subinterval

without being the null function. Just for that reason zeros are counted

according to the rules as given in Definitions 1.4.9 and 1.4.10. We reeall

that the multiplicity of a zero of a function depends on the choice of the

smoothness class in which the function is considered. Given an arbitrary

!-spline s of order n defined on a finite interval [a,b], one always has

s E PC([a,b]). Between consecutive knots the function s coincides with a

function in Ker(pn) and since the number of knots in (a,b) is finite, it

fellows that the total number of strong zeros of s in (a,b) is also finite.

In what fellows we give in particular attention to !-spline functions with

simple knots. To begin with, Corollary1.4.12 yields the following

LEMMA 1.4.21. Lets be an l-spZine function on [a,b] of order n with simpZe

knots. If the corresponding operator p (D) is disconjugate on [a,b], then n

(cf. pp. 11-14 for notation)

[n-1] D1 and D = Dn-l ••• D1 (cf. (1.4.8)),

[n-lJ (n 1) PROOF. We note that D s E PC - ([a,b]) and thus Corollary 1.4.12 may

be applied with n replaced by n-1. D

. [ n-1] [ n-1] The funct~on D s satisfies the equation pn(D)s = DnD s = 0 between

each two consecutive knots in (a,b), hence (cf. (1.4.2)) D[n-l]s(t) - [n-1]

= c wn (t). As a consequence we have S (D s, (a,b)) "; k, where k is the

number of knots in (a,b). This observation combined with Lemma 1.4.21 gives

the following

COROLLARY 1.4.22. Let s be an l-spZine on [a,b] of order n with simpZe

knots. If the corresponding operator pn(D) is disconjugate on [a,b], then

n-1 Z ((Di)

1 ,s, (a,b)) "; k + n- 1 ,

where kis the number of knots of sin (a,b).

24

The Budan-Fourier theerem 1.4.14 applied to t-splines yields

THEOREM 1.4.23. Lets be an t-sptine funation on [a,b] of order n with

simpte knots. If the aorreaponding operator pn(D) is disconjugate on [a,b] • [n-1] [n-1] and ~f D s(a)D s(b-l ~ 0~ then

- [n-1] - [1] [n-1] :SS(D s,(a,b))+S(s(a),D s(a), ••• ,D s(a))+

+ [ 1] [n-1] -s (s(b),D s(b), ... ,D s(b-)),

[i] where D (i= 1,2, ••• ,n-1) ia given by {1.4.8).

COROLLARY 1.4.24. Let s be an t-apUne on [a,b] of order n with simpte

knots. If the aarreeponding operator pn(D) ia diaaonjugate on [a,b] and if D[n-l]s(a)D[n-l]s(b-l ~ 0~ then

- [1] [n-1] Sk+S (s(a),D s(a), ••• ,D s(a)) +

+ [ 1] [n-1] - s (s(b),D s(b), ••• ,D s(b-)) ,

where k ia the nwrber of knots of s in (a,b).

1.4.7. The operator p*(D) n

The operator p*(D) is defined by n

If ~is the fundamental function oorreeponding to the operator pn(D), then n-1 evidently t a...-..t (-1) ~ (-t) is the fundamental function corresponding to

* the operator p (D). Furthermore, we note that intervals of disconjugacy n * for pn(D) arealso intervals of disconjugacy for pn(D), as

* (t....., f(-t) J E Ker(pn) whenever f E Ker(pn).

25

1.4.8. Divided differences

Let n + 1 points x0 :S x1 :S ••• :S xn with x0 < xn be given and let [x0 ,xn] be

an interval of disconjugacy for the operator pn(D). Due to Lemma 1.4.17 the

set V c Rn+l defined by

n+1 has dimension exactly n, and therefore there exists a vector ~ E R ,

uniquely determined up to a multiplicative constant, such that

(1.4.25)

To ensure the uniqueness of a a further condition is needed. Hence, we

require that

(1.4.26) 1 ,

where gis a function satisfying pn(D)g{t) n~(tER).

DEFINITION 1.4.25. If ~ E Rn+ 1 is the unique vector satisfying (1.4.25} and

(1.4.26), then the expression !<x0 ,x1 , ••• ,xn) is caZZed the divided

difference f with respect to the points x0 ,x1 , ••• ,xn and the operator

pn(D); it is denoted by f[pn;x0 ,x1 , ••• ,xn].

n In the particular case that pn(D) = D we obtain the ordinary divided dif-

ference common in numerical analysis and for this the notation

f[x0 ,x1, ••• ,xn] will be used.

It fellows from (1.4.25) and (1.4.26) that

<p <p' Cjl (n-1)

f

xo xl x n-1 x

(1.4.27) f[pn;xO,xl, ..• ,xn] n

(n-1) <p <p' <p g

xo xl x n-1

x n

n If pn(D) = D and x0 < x 1 < ••• < xn, then formula (1.4.27) reduces to the

well-known formula (cf. Oavis [ 16, p. 40])

26

where wis defined by W(t) := (t-x0) ••• (t-xn) (t ":R).

T . Writing ~ = (a0 ,a1 , ••• ,an) , we may compute the numbers ai from formula

(1.4.27) by expanding the determinant in the numerator of (1.4.27) with

respect to its last column. One then obtains

!p lP' lP (n-1}

(-1) n+i xo xl xi-1 xi+1 x

( 1. 4. 29) n ai (n-1} •

(j) q>' q> g

xo x1 x n-1 x n

According to Example 1 on p. 19, the functions , (n-1) q>,q> , ••• ,<p form an ex-

tended Chebyshev system of ordernon [x0 ,xn]. This implies that the deter­

minant in the numerator of (1.4.29) has constant sign, hence

2. THE B-SPLINE FUNCTIONS

2 • 1 • 1 rttttodu.c.tio n a.nd .6 wnma.Jty

The subject of this chapter is the class of the so-called basic t-spline

functions, B-spline functions or B-splines for short. This name finds its

origin in the fact that for some spaces of l-splines these functions form

a basis; e.g. in the case of polynomial splines the well-known polynomial

B-splines form a basis. For various properties of:polynomial B-splines the

reader is referred to Lectures 1 and 2 of Schoenherg's monograph [51].

Fundamental properties of B-splines corresponding to a general operator

pn(D) are studied in a paper by Schmidt, Lancaster and Watkins [50].

27

For the sake of completeness, in Sectien 2.2 we prove some fundamental

properties of B-splines. Sectien 2. 3 is concerned wi th the se-called total

positivity property of a consecutive sequence of B-splines. Applying our

version of the Budan-Fourier theerem to l-splines (Theorem 1.4.23) we give

a rather simple proof of this property. Using the Fourier transferm in

Sectien 2.4, we derive various recurrence relations for B-splines.

In the final Sectien 2.5 one of these recurrence relations is used to in­

vestigate the behaviour of the supremum norm of the polynomial B-spline of

order n as a function of its knot distribution. The optimal distribution of

the knots, i.e., the distribution for which the supremum norm is minimal,

is determined; this answers a question of Meinardus [34, p. 174].

2.2. Funda.mental-6 a.bout B-.6ptine.6

2.2.1. Definition of a B-spline function

The B-spline will be defined by using the concept of divided differences as

given in Definition 1.4.25. In order to obtain B-splines'corresponding to

the operator pn(D) weneed divided differences with respect to the operator

* pn(D) (cf. Subsectien 1.4.7).

28

Let x0 ~ x1 ~ ••• ~ xn be a sequence of points wi~ x0 < xn and let [x0 ,xn]

be an interval of disconjugacy for the operator p (D) • Since * * n f[p ;x0,x1, ••. ,x J = 0 for all f E Ker(p), it fellows from Lemma 1.4.5 n n n

(Peano's remainder formula) that a function K exists such that

(2.2.1)

xn

I K(t)p:(D)f(t)dt

xo

* The kernelK is given by K(T) := f 0[pn;x0 ,x1 , ••• ,xn], where f0

is defined

* * by f0

(t) := cp+(t-T) and cp+ is the Green's function corresponding to the

* operator pn(D) * (cf. Definition 1.3.3). * The function cp+ can be expreseed in terms of cp+ and cp by means of

(2.2.2) (t E lR)

* * * Since cp [pn;x0

,x1 , ••• ,xn] = 0 it fellows that K(T) = f 1[pn;x0 ,x1 , ••• ,xn]

wi th f 1

( tl : = ( -1) n cp + ( T - tl •

Using thesepreliminaries we arrive at the following definition.

DEFINITION 2.2.1. Let x0 ~ x1 $ ••• $ xn be a sequenae of points with

x0 < xn and "let [x0 ,xn] be an interval- of disaonjugaay for the ope:r>ator

pn{D). The B-sp"line funation M(pn;x0 ,x1 , ••• ,xn;•) of o!'der n ao:r>responding

to the operatoP pn(D) and having knots x0

,x1

, ••• ,xn is fortE lR de~ned by

(2.2.3)

On account of this definition and FOrmula (1.4.27) it is easily verified

(cf. (1.3.1)) that M(pn;x0 ,x1 , ••• ,xn;•) is indeed an l-spline function of

order n corresponding to the operator pn(D) and having the knots

xO,x1' • • .,xn.

In accordance with Definition 2.2.1, the B-spline corresponding to the

operator p (D) and having knots x.,xj 1

, ••• ,xj will be denoted by n J + +n

M(pn;xj,xj+i'••••xj+n;•). As B-splines will be heavily used in this

chapter, it is convenient to have other notations available that are more

.. :t01i1111fact. We define

M. (p • •) J n'

and in case of polynomial B-splines of order n we introduce the notatien

The B-spline M0 (pn;•) with equally spaeed knots xi = ih (i= O,l, ••• ,n)

where h is a positive number is denoted by Bh(pn;•), and in the case of

polynomial B-splines of order n we introduce the notatien

Bn,h := M(Dn;O,h, •.• ,nh;•). Obviously one has

If

or

M(pn;jh, ••• , (n+j)h;t)

p (D) n

p (D) n

2 2 2 2 2 D (D + w I) • • • (D + m w I)

2 1 2 2 9 2 (D +4 w I) (D +4 w I)

(t E: JR) •

(n 2m + 1 w ~ 0)

(n = 2m ; w ~ 0) ,

the corresponding B-splines are called trigonometrie B-splines.

If

or

p (D) n

2 2 D (D - w I) • • •

2 2 - m w I)

2 1 2 2 9 2 (D - 4 w I)(D - 4 w I)

(n = 2m + 1 w "' 0)

(n = 2m ; w ~ 0) ,

the corresponding B-splines are called hyperbalie B-splines.

29

In the case of polynomial B-splines of order n we have pn(D) Dn and (cf. 1 n-1

Definition 1.3.3) ~(t) = t , hence

(2.2.4)

Furthermore, tf x0

< x 1 < ••• < xn then according to (1.4.28) Formula

(2.2.4) may be written as

(2. 2 .5) n

M 0

<tl = (-1ln n L n, j=O

where W (t) := (t- x0

) • • • (t- xn). This representation precisely agrees with

Formula 2,7 in Schoenberg [51, p. 2].

30

2.2.2. Some general properties of B-splines

We begin this subsectien with a representation formula for M0

(pn;•). For

that purpose the distinct points among the knots x0 s x1 s .•. s xn are

denoted by y0 < y 1 < ••• < yk, their multiplicities by p0 ,~ 1 , ••• ,~k'

respectively. It then follows from (2.2.3) 1 (1.4.29) 1 and (1.4.30) that

(2.2.6) (t € :R) I

where the coefficients W0 • when ordered as w0 0

,w0 1

, ••. ,w0 1

, "1 J 1 , , P0-

w1 0

, ••• ,w1 _1, •.• ,wk 0 , ••• ,wk 1, alternate in sign and are uniquely I 1 ~1 I 1 ~k-

determined,

In Schoenberg[Sl,pp. 2,3]it is shown that the polynomial B-spline M 0 has n, the properties M o<tl > 0 if xo < t < x , M o<tl = 0 if t < xo or t >x I .., n, n n, n and J M

0 (T} dT = 1. In what fellows we prove that the B-spline M

0 (p ; •) - n, n

has the same properties.

As an immediate consequence of (2.2.1), (2.2.2),.and (2.2.3) B-splines

satisfy the relation

(2.2. 7) 1 =-

n!

xn

f MO{pn;T}p~{D)f{T)dT •

xo

Applying {2.2.7) toa function g for which p~(D)g(t) using (1.4.26) we obtain

n! (t e :R) and

(2.2.8) xn

f M0 {pn;T)dT = 1.

xo

n-1 * As q>+(t-T) = 0 (T ~ t), ip+(t-T) = ip(t-T) = {-1) ip {T-t) and

* * lP (• -tl[pn;x0 ,x1, ... ,xn] = 0, Formula (2.2.3) implies that

(2.2.9)

Thus a B-spline has compact support, and in view of (2.2.8)

00

J M0 Cp0

;<ld-r 1 •

-co

31

We praeeed by showing that B-splines are positive on their supports, i.e.,

that

In the case of simple knots this can be established by using the generalized

Budan-Fourier theerem 1.4.23 for .C-spline functions (cf. Lemma 2.3.1, p. 34);

however in the case of multiple knots we need Lemma 1. 4. 19 to der i ve

(2.2.10). A proef of (2.2.10) for the general case will now be given.

PROOF. Since pn(D) is disconjugate on [x0 ,xn] and since the maximal inter­

vals of disconjugacy are open, a number b > xn exists such that pn(D) is

disconjugate on Cx0 ,b). Now assume that M0 (pn;t0J 0 forsome t 0 E (x0 ,xn).

Choose n arbitrary points t 1 < t 2 < ••• < tn in (xn,b) and define xi+n := b

(i 1,2, •.• ,n). Then M0 (pn;ti) = 0 (i= O,l, ... ,n). In view of (2.2.6) we

have

(2.2.11) 0 1

where the functions ~i are defined as in Lemma 1.4.19. However, since

xi <ti< xi+n (i O,l, ••. ,n) it fellows from Lemma 1.4.19 that the

determinant in (2.2.11) is positive; this contradietien proves that M0 (pn;')

has no zeros in (x0 ,xn). It then fellows from (2.2.8) and the continuity of

REMARK. From (2.2.8), (2.2.9), and (2.2.10) it fellows that a B-spline is a

probability density on (x0 ,xn). For probabilistic interpretations in the

polynomial case we refer to Feller [17, p. 28].

2.2.3. The basic property of B-splines

Whereas Formula (1.3.1) represents an .C-spline in termsof the Green's

function ~+' one may also show that B-splines can be used to represent an

arbitrary .C-spline. This property of B-splines is the content of the

following

THEOREM 2.2.2. Let the sequence of knots (xv)=~ satisfy xv ~ xv+1,

x < x + (v € 'JZ.), lim x = "" and lim x "' - "" • FurtheY'I11ore, Zet the V V n ~ V v+-~ V

0

32

operatoP pn (D) be diaconjugate on eaah inte'!'Vat [ xv ,xv+n] (v .: li:). Then foP

every l-sptine s coP'l'eaponding to the ope'l'ato'l' pn (D) and having knots

(xv>:w the'l'e exiata a unique aequence (cv)~ of Peal numbe~ auch that

(2.2.12) s(t) (t E: JR) •

PROOF. Withoutlossof generality we may assume that x0

< x1

• We first show

that the B-splines Mv(pn;•) (v=l-n, ••• ,O), when restricted to the interval

(x0 ,x1) form a basis for Ker(pn). For this it is sufficient to prove that

thesen functions are linearly independent on (x0

,x1). For this purpose let

us assume that a linear combination

(2.2.13) f(t} 0 I À M (p ;t)

v=l-n v v n

vanishes identically on (x0 ,x1). The function f is an !-spline with knots

x1_n,x2_n'''''x0 ,x1 , ••• ,xn. According to the representation (1.3.1) for f

in terros of q1 , there exists a vector d € JRn such that + -

Since f(t) = dT q~(t-x 1 ,t-x2

, ••• ,t-x0

) = 0 (x0

< t < x1

) it follows from - - -n -n

Lemma 1.4.17 that ~ = Q· Hence f(t) 0 for all t.: [x1_n,x1}. Formula

(2.2.13) tagether with (2.2.9) implies that one successively bas Àl-n =

= Ä2-n = ••• = Ào = o. Consequently, there exists a unique sequence of numbers c 1_n,c2_n, ••• ,c0 such that

0 s(t) := s

0(t) := I cM (p ;t)

v=l-n v v n

He nee s- s 0 is an .C-spline wi th knots (x ) "" , which is identically zero on \)-<0

(x0 ,x1), and thus s-s0 can be written uniquely as s-s0 = s 1 +s 2, where s 1 and s 2 vanish on (-~,x 1 ) and (x0 ,oo), respectively. If x1 has multiplicity

r, then by (1.3.1) s 1 is a linear combination of the functions

t~qJ(i) (t-x1) (i= O,l, ••• ,r-1) on the interval (x1,xr+l) and thus in

view of (2.2.6) s 1 can be written uniquely as a linear combination of the

functions Mv(pn;•) (v = 1,2, ••• ,r}. Repeated application of this argument

leads to the unique representation

L c M (p ;t) v=1 v v n

(t E JR) •

In a similar way it can be shown that s 2 can be represented uniquely as

-n L cvMv(pn;t) (t E JR) •

V=-oo

As s s 0 + s 1 + s2

this establishes the theorem.

Given an operator pn(D) that is disconjugate on JR and given a sequence of

simple knots (x )00

(V E ZZ), a result of Karlin [21, Theorem 4.1, p. 527] V -"'

implies that the corresponding sequence of B-splines (M (p ;•))00

(cf. v n V=-"'

p. 28 for notation) has the property

M (pn; •) V

(2. 3 .1) r

~ 0

t;1 t; r

33

D

for all t;0 < t; 1 < ••• < sr' v 0 < v1 < ••• < vr (rE N0

); we reeall that the

determinant in (2.3.1) is defined in Subsection 1.2.6.

Property (2.3.1) is known as the totaZ positivity property of the sequence

of B-splines (Mv(pn;•))~=-"'· An important consequence of the total positivity property of the sequence

(M (p ;•))00

is contained in Theorem 1.3 (cf. Karlin [21, p. 531], which v n V=-"'

can be statedas follows: if an !-spline s can be represented in the form

s(t) = t:"' cM (p ;t), then S-(s,lR) ~ S-((c )00

); this is known as the v=-"' v v n v -"'

variation diminishing property of B-splines.

With respect to the required disconjugacy condition for the operator pn(D)

we reeall (cf. Section 1.4) that pn(D) is disconjugate on JR if and only if

all zeros of pn are real. In view of this it follows that in particular the

sequence of polynomial B-splines (M )00

with simple knots is totally n,v v=-00

positive.

Using Theorem 1.4.23 (the Budan-Fourier theorem for !-splines) we shall

give a rather simpleproof of (2.3.1) under the stronger condition that

34

<v 0 ,v 1, ••• ,vr) is asequenceofaonseeutive integers {V,v+l, ••• ,v+r). In

the course of the proof the following lemma is needed.

LEMMA 2.3.1. Let r E 1'10 and Zet x < x +l < ... < x + be suah that V V V n+r

[ xv, xv+n+r] is an inte'!'Val of disaonJugaay for the operator pn'(D). Then for

every sequenae of real nUI'Tbers a 0 ,a 1 , ••• ,ar the nUI'Tber of strong zerosof

E~=Oaj Mv+j(pnp)~ aounting rrrultipliaities aaaording to Definition 1.4.10"

is at most r" i.e. (af. pp. 14 for notation)

(2.3.2)

û.lhere Pn (D) Dn ••• D1 (af. (1.4.1). (1.4.2)).

PROOF. The proof will be given by induction with respect to the variable r.

Let fO := M (p ;•) and for r ~ 1 let f be defined by f := I:rj-..1'\a..MV+j(p ;•). v n r r -v J n Inequality {2.3.2) in the case r = 0 asserts that Mv(pn;•) has no strong

zerosin (xv,xv+n). Using Lemma 1.4.19 we have shown this to be true (cf.

(2.2.10)) for knots with arbitrary multiplicity. However, in the case of

simple knots the Budan-Fourier theorem for l-splines can be used to prove

that f0

> 0 on (x ,x ), since then f0

E PC(n-l) ([x ,x ]). It follows v v+n v v+n

from (2.2.6} that f 0 can be written in the form

(2.3.3)

where

(2.3.4)

n

l wR.cp+(t-xv+R.) R.=O

(t E JR) 1

In view of Definitions 1.3.4 and 2.2.1 one has

(i = 0,1, ... ,n-2) •

Taking into account Definition 1.3.3 and (2.3.3) together with (2.3.4), we [n-1] n

conclude that D f 0 (xv) = w0 f: 0. As E R.=O wR. cp (t- xv+R.} = 0 (t E :R} it

follows that

D[n-l]f (x -) 0 v+n

the last inequality again being a consequence of (2.3.4). Applying Theorem

35

1. 4. 23 we get

s n- 1- (n-1) = 0 ,

- [ n-1] as an upper bound for S (D f 0 , (x\1, xv+n) ) is gi ven by the number of knots

in (x\l,xv+n) (cf. Corollary 1.4.24). This proves (2,3.2) for r 0. We

preeeed by an induction argument with respect to r. In view of (2.2.9), fr

is identically zero outside the interval [x ,x ]. If a0

0 or a 0, \i \l+n+r r

then by the inducti.on hypothesis we may conclude that n-1

Z ((Di) 1 , (x ,x ) ) s r- 1 < r. If, however, a 0 'I 0 and ar 'I 0, then \i \l+n+r[n-1]

0 [ n-1] (x\1) 'I 0 and D fr(x\1+-n+r) 'I 0, and thus according to Corollary

1.4. 24

This completely proves the lemma. D

Next we shall prove (2.3.1) under the assumption that (v0

,v 1 , ... ,vr) is a

sequence of consecutive integers (v,v+l, ••. ,v+r).

THEOREM 2.3.2. Let p E rr (n ~ 2) be a monic polynomiat having only reaZ n n zeros and Zet (x )"' be a sequenee of simpte knots. Then for every r É ~O

\) -<X>

and for every aequenee of numbers ~ 0 , ••• ,~r with ~O < ~ 1 < ••• <~rand

every v E 7Z,

(2. 3.5) ~r

~ 0 .

Moreover, strict inequaZity in (2.3.5) prevaiZs if and onZy if

(2. 3 .6) (i= 0,1, ... ,r) •

PROOF. As a preliminary remark we want to emphasize that throughout the

proof Formula (2.2.9), i.e., M\l(pn;t)

role.

0 fort~ (x ,x+), plays a crucial v \i n

If forsome 10 E {O,l, ••• ,r} (2.3.6) does not hold, then either ~. s x. J.o J.o+v

or ~. ~ x. and thus in view of (2.2.9) J.o J.o+v+n

36

0 (j o,t, ... ,i0l

or

= ••• 0

Therefore, if t;io ~ xio+\1 it is easily seen that the first io + 1 columns

in the determinant of (2.3.5) are dependent. Hence the determinant is zero;

in the other case a similar argument applies. Consequently, (2.3.5) is

established if (2.3.6) does not hold.

There remains to be proved that the determinant is positive if (2.3.6)

holds. This will be done by induction with respect to the variable r.

If r = 0 then (2.3.5) and (2.3.6) assert that the B-spline Mv(pn;•) is

positive on (xv,xv+n)' which is obviously true in view of (2.2.10).

We proceed by defining the function f as

Mv (pn;t) Mv+r(pn;t)

Mv (pn;t;1l Mv+r (pn;!;: 1) (2.3.7) f(t) := (t € JR)

Mv (pn; !;:r) Mv+r (pn; t;r)

where (cf. (2.3.6)) xi+v < t;i < xi+v+n (i= 1,2, ••• ,r).

In view of this definition and (2.3.5) one has to prove that f(t; 0) > 0 for

all ~;: 0 < t; 1 and t; 0 € (xv,xv+n). It follows from (2.3.7) that

f(t) E;=OajMv+j(pn;t) and f(l;i) = 0 (i= 1,2, ••• ,r). Now suppose that

also f(t;0

l = 0. As pn has only real zeros, it follows that pn(D) is dis­

conjugate onlR. Hence Lemma 2.3.1 is applicable and one has n

Z((Dil 1,f,(xv,xv+n+r)) ~ r. Consequently, at least one of the zeros

t;0 , ••• ,t;r.is not strong. Let ~i 1 be the smallest zero among t;0 , ••• ,t;r

which is not strong. Then by the definition of a strong zero (cf. p. 12)

f vanishes identically between two consecutive knots ~-l and ~ with

t;11

€ [~_ 1 ,~], since fis an l-spline function. It fellows from the proef

of Lemma 2.1.2 that for t € <-~,~~

(2. 3.8) f (t)

{

k-1-n-v L a. Mv+j {pn; t)

j=O J

0

(k ~ v+n+1) ,

(k < v+n+l) •

37

If k ~ v+n+l, then in view n-1 of (2.3.2) one has Z((Di)l ,f,(xv,xk)) k-1-n-v.

Since, moreover, x. < ~. J.o+v J.l

< xi and ~. c [ xk 1, x.] one has 1 +v+n J.1 - .K

k-1-n-v s i 1 +v+n-1-n-v =i 1-1, and again one has to conclude that at least one

of the zeros ~ 0 ,~ 1 , .•• ,~i 1 _ 1 is not streng. This, however, contradiets the

assumption that ~il is the smallest zero among ~O'''''~r which is not

strong, and thus k ~ v+n+l does not hold. Consequently, k < v+n+l and there-

fore by (2,3.8) f vanishes identically on (-"",xk). Since ~il E (xi1+v'xi

1+v+n)

and ~il E [ ,~] one has k-1 ~ i1+v :?. v. Hence k ~ v+l and f vanishes

identically on (xv,xv+l). However, taking into account (2.3.8) and (2.2.9),

on (xv,xv+l) we may write

f(t)

which by the induction hypothesis and (2.2.10) is strictly positive on

(xv,xv+l). This contradiets the fact that fis identically zero on

(xv,xv+l). So f(~0 ) # 0 and it remains to prove that f(Ç 0) > 0. This can be

done as fellows. Since n ~ 2 and all knots are simple the B-splines Mv(pn; )

(v c ~) are continuous on R. Therefore, the determinant in (2.3.5) is con-. T r+l

tinuous wJ.th respect to (ç0 ,~ 1 , .•. ,~r) ER 1 and soit doesnotchange

sign on the open and connected set {xi+v <~i< xi+n+v i O,l, ••• ,r}, By

taking the specific contiguration ~i (xi+v'xi+l+v), we see that (2.3.5)

is equal to n~=O Mv+i (pn;Çi) > o. D

REMARK 1 • If n 1, then a straightforward calculation shows that for all

:?. 0

for all ~O < ~ 1 < ••• < ~r with strict inequality if and only if

xi+v s ~i< xi+v+l (i O,l, ... ,r).

REMARK 2. If not all zeros of pn are real then an additional condition for \)=»

the knot sequence (xv) is neededinorder to obtain (2.3.5). We note

that for r = 0 Theorem 2.3.2 reduces to property (2.2.10). There we made

the assumption that pn(D) is disconjugate on [x ,x+~. It is an open v v n

38

problem whether the assumption that pn (D) is disconjugate on ~xv ,xv+n]

(v E: lZ;) guarantees ( 2. 3. 5) for all r ~ 1.

2.4.1. The Fourier transferm of a B-spline

Given the B-spline Mv(pn;•) with knots xv,xv+l'''''xv+n' its Fourier trans­

form is defined as

00

(2.4.1) I -Z'r ~ (z) := Mv(pn;-r}e d-r n,v (z € 4::> •

In view of (2.2.8) one obviously has

(2.4.2) ~ (0) = 1 • n,v

LEMMA 2.4.1. Let~ be the Fourier transfarm of Mv{pn;•l, then the fu:na-n,v tion H defined by n,v

(2.4.3} H (z) := p (z)~ (z} n,v n n,v (z E t:>

has the property

(2.4.4) (D +x" I) (D + x"+li) • • • (D +x I)H (z} = 0 v v v+n n, v (z € (:) •

PROOF. As a consequence of (2.4.1) one has

H (z) n,v

and thus in view of (2.2.7) and (1.4.27) we conclude that (2,4.4) holds.

REMARK. For polynomial B-splines (n)

(2.4.3) and (2.4.2) imply that H(i) (0) n,v (i= O,l, ••• ,n-1), H (0) = n!. n,v Hence, as a consequence of (2.4.4),

H /n~ is the fundamental function corresponding to the operator n,v (D+x I) (D+x +lil ••• (D+x ..... I) (cf. Cox [15]; Ter Morsche [42, p.

'\) '\) v<n ' 5]).

0

0

39

2.4.2. Some recurrence relations for B-splines

The first of a series of recurrence relations for B-splines is contained

in the following

LEMMA 2.4.2. Let x0 x 1 ~ ••• ~ xn+l be a sequence of knots with x0

< xn

and x 1 < xn+l' Let À E ~ be such that pn+l(D) and pn(D) satisfy the condi­

tions

i) pn+l (D) = (D- ÀI) pn (D) ,

ii) pn(D) is disconjugate on both [x0 ,xn] and [x1 ,xn+l], and pn+l(D) is

disconjugate on [x0 ,xn+l].

Then one has the recurrence relation

(2.4.5) (D- ÀI) M0 (pn+l ;tl (tE~) 1

where

(2.4.6) {

H o(À)(<P l(À) <P o(À)J-1 n, n, n,

a= (~' (0) ~· (0))-1 n,l n,O

(À ,; 0) 1

(À 0) •

PROOF. The function (D ÀI)M0

(pn+l;•) is an !-spline corresponding to the

operator pn(D) and having the knots x0 , ... ,xn+l' Because of (2.2.9) it is

identically zero outside [x0 ,xn+l], and in view of Lemma 2.2.2 it can be

written as a linear combination of the two B-splines M0 (pn;•) and M1 (pn;•).

In order to obtain the coefficients in this linear combination we take the

Fourier transferm of both sides in (2.4.5) yielding (as <P (0) n,v n E N and v E 1Z the coefficients must sum to -À)

(2.4. 7) (z-À)~n+l,O(z) a<P 1(z)- (a+À)<P o(z) n, n,

Letting z + À gives (2.4.6) and inversion yields (2.4.5).

1 for

For polynomial B-splines, i.e., when pn+l (D)

(2.4.5) and (2.4.6) take the form

Dn+ 1 , formulas

(2.4.8) (M O (t) - M l (t)) xn+1 - x0 n, n,

+ 1 (t ,E ~) ,

In order to verify (2.4.8) we compute a as given by the secend formula of

(2.4.6). Tpis involves the evaluation of <P' (0), which can be performed n,v

D

40

as fellows, In view of (2.4.3)

on p. 38) that

z0 <I> (z), one bas. (cf. remark n,v

H(i) (0) = Q n,v (i= O,l, ••• ,n-1) , H(n) (0)

n,v n: ,

and

H(n+l) {0) = (n + 1) ~ <I>' (0) n,v n,v

On the ether hand, (2.2.4) implies that

Consequently,

!j)' 0(0) -1 ( ) n, = n + 1 xO + xl + '• • + xn

and thus CL =

For numerical purposes the recurrence relation

(2.4.9) 1 __ o M (t) + n+l M (t) • (t-x x -t )

xn+l -x0 n n,O n n,l

where t E R, is of importance, as it is useful to compute B-splines in a

stable manner (cf, Cox [15]).

It turns out that (2.4.9) may be derived from Theerem 2.4.3 {cf. p. 42).

The Fourier transferm can be a useful tool to construct formulas similar to

(2,4.9) for B-splines corresponding to specific operators p0

(D). For in­

stance, in the case of trigonometrie B-splines (cf. p. 29) a formula similar

to (2.4.9) is due toKoch and Lyche [26, p. 188].

As an illustration, we here derive a recurrence relation for hyperbalie

B-splines {cf. p. 29). So, let n = 2m (m E ~) and let w E ~ with w ; 0. The

operators pn(D) and p0

+1

(D) are defined by

m 2 2 2 ;= 0 (D - {j- il w I) ,

j=1

Pn+l (D) m 2 2 .

;= D 0 (D - (jw) I) • j=l

For simplicity we assume the knots x0 ,x1 , ••• ,xn+l to be simple. We reeall

(cf. (2.4.1)) that the Fourier transferros of M0 (pn+l;•), M0 (pn;•) 1 and

M1 (pn;•) are denoted by ~n+l 1 01 ~n,O and ~n,l' respectively. Furthermore,

let Hn+l,O' Hn10' Hn,l' according to (2.4.3), be the functions related to

~n+l,O' ~n,O' ~n,l' Now constants a 1 and a 2 exist such that

(2.4.10)

the existence of these constants is guaranteed, because H (z- w and n,O Hn,l (z ~) satisfy (2.4.4) with \1 = 0, and n replaced by n + 1. Hence by

(2.4.3) Hn,O(z-~) andHn,l(z-~) vanish at the zerosof pn(z-~), which

polynomial is a divisor of pn+l' Consequently, an appropriate linear com­

bination of H 0 (z-~2 J and H 1 (z-~2 J vanishes at the zerosof p 1

• n, n, n+ We note that the coefficients bi, bi,O and bi, 1 in the expansions

r "n+!,o'"'

(2.4.11) l Hn,O (z)

H 1

(z) n,

-x z n

are even functions of w. Substituting (2.4.11) into (2.4.10) and equating -x.z

the coefficients of e ~ 1 we obtain

where

Multiplying (2.4.10) with pn(z-~) and using the relations pn+l(z) w (z+mw)pn(z-2) and (2.4.3), we get

(2.4.12) -x0w;2 w -xnw/2 'W

c1

(w)e ~ (z--) + c2

(w)e ~ 1<z--2J n,O 2 n,

(z+mw)~ 1 o<zl • n+ ,

42

Replacing w by - w yields the identi ty

(2.4.13)

= (z- mw) ~ 1 0

(z) n+ ,

The required recurrence relation is now obtained by subtracting (2.4.13}

from (2.4.12) and then taking Fourier inverses. For the hyperbolic B-splines

under consideration this gives rise to the following

THEOREM 2 .4. 3.

(2.4.14)

(t f. R) •

In order to apply (2.4.14) the coefficients c 1(w) and c2 (w) must be evalu­

ated explicitly. Furthermore, we note that (2.4.9) may be obtained from

(2.4.14) by letting w tend to zero.

Our next purpose is to deri ve yet another recurrence relation for polynomial

B-splines, which will be needed in the following section.

LEMMA 2.4.4. Let M 0

be the poZynomiaZ B-sp'line of order n wi,th knots n,

x0 ,x1, ••• ,xn. Let x be an arbitrary point in [x0 ,xn] and Zet i f. {1,2, ••• ,n}

be aueh that x E [x1

_ 1 ,x1

J. n+1 ,

Pu:r>thermore, Zet Mn+1,(xl := M(D ;x0 , ••• ,x1_ 1,x,x1 , ••• ,xn;·) be the poZy-

nomiaZ B-ap'line of order n+l ü)ith knots x0 , ... ,x1

_1,x,x1 , ••• ,xn. Then

(2.4.15) Mn+l,(x)(t)=t-xM' (t)+n+lM O(t) n n+ 1, (x) n n, (t f. R) •

PROOF. The fundamental functions eerrasponding to the operators

and

are denoted by ~n+l and ~n' respectively. The Fourier transfarms ~n+l,(x)

and <l>n10 of Mn+ 11 (x) and Mn

10 are given by the relations (cf. (2.4.3) and

the remark on p. 38)

n+1 z (n+1)~ <l>n+1

1(x)(z) 1

n z

cpn(z) =I <I> O(z) n. n1

The functions cpn+ 1 and cpnare related by (D +xi)cpn+1 cpn. Hence

(2.4.16) <I> (z) = ~ <!>' (z) -~<I> +<I> o<zl n+1 1 (x) n+1 n+1 1 (x) n+1 n+1 1 (X) n 1

Inversion of (2.4.16) yields (2.4.15).

2.4.3. B-splines with equidistant knots

For equidistant knots 0 1 h 1 ••• 1 nh (h > 0) the Fourier transferm of the B­

spline function Bh(pn;•) may be easily obtained as fellows. If

p (z) = 11~ 1 (z- S .) (S. E tl then 1 according to (2.4.3) and (2.4.4) 1

n J= J J H

0(z) is a scalar multiple of the function

nl

z + n -hS. n (e -hz - e J)

j=1

The function <I> is then given by n 1 0

(2.4.17) <I> o<zl = d n 1 n

n n

j=1

where d is a constant such that <I> 0 (0) = 1 (cf. (2.4.2)). n n,

It fellows from (2.4.17) that Bh(pn;•) equals the convolution product

f 1 * f 2 * ... * fn 1 where

-l Sih -1 Sit

si (e - 1) e (0 $ t < h; si -1 Ol I

(2.4.18) f. (t) -1

(0 s. = 0) h $ t < h; I ~ ~

0 (t < 0 or t 2: h) \

We cbserve that (2.4.5) and (2.4.6) may also be simplified in the case of

equidistant knots. For this purpose we introduce the ,shift operator E.

43

D

44

DEFINITION 2.4.5. Given a positive number h~ the shift operat?r E is defined

as fotloi;;s. For eaah real-valued funation f defined on :R the funation Ef is

given by

Ef(t) := f(t +h) (t E :R) •

0 n n-1 As usual, E := I and E := E(E ) (n E E).

Now let pn+l (z) = (z-À)pn(z) with À E :R. Taking into account (2.4.17) one

has

Inversion yields the identity

( t E :R) •

If À= 0, then À(l-e-Àh)- 1 has to be replaced by h-1 For polynomial B­

splines with h = 1 recurrence relation (2.4.19) takes the simple form

(2.4.20) B~+l,h{t) = (E-I)Bn,h(t-1),

which relation is contained in Ter Morsche [41, p. 211].

2. 5. The. müumwn .&u.pJtemwn noJUn o6 a. polynomé..ai. B-.&p.Une.

2.5.1. The normalized polynomial B-spline basis

Let s be a polynomial spline of order n defined on [a,b] and having the

knots y 1 < y 2 < y 3 < ••• < yk with multiplicities ~ 1 .~ 2 , ••• ,~k' respective­

ly. We reeall that s can be represented by the so-called truncated power

basis (cf. (1.3.1)), where in the case of polynomial splines ql+(t) = n-1 = t+ /(n- 1)!. In order to express the function s as a linear combination

of polynomial B-splines the points a and b are considered to .be knots of

multiplicity n, so we introduce additional knots x0 = x 1 = ..• = x0

_ 1 =a

and x.t+l = xJ1,+2 = ••• = x.t+n = b, where t = n + ~~ + ••• + ~k- 1.

If we put x = x +l = ... = x + 1 =y1 , x + • x + +1 n n n ~~- n ~1 n ~1 x =y, ••. ,x =x = n+vt+~2-1 2 n+~t+ ••• +Vk-1 n+~t+ ••• +vk-1+1

=x+ 1

= yk, the function scan (cf. (2.2.12)) uniquely be repre-n ~1+. • .+Vk-

sented as

45

R, s (t) ~ C, M . (t)

i=O ~ n,~ (a < t < bl •

Hence the polynomial B-splines M . (i O,l, ••• ,R,) forma basis for the set n,~

of all polynomial splines of order n with fixed knots y1

, ... ,yk with multi-

plicities v1,v2 , .•. ,vk, respectively.

As a slightly different basis for the polynomial spline functions one aften

uses the so-called normalized polynomial B-splines N . := N . (x., ••. ,x,+n; n,~ n,~ ~ ~

(cf. De Boor [3]), which are related to the polynomial B-splines M . by n,~

(2.5.1) N . (t} = ~+n ~ M , (t) (x. -x.) n,~ n n,~

(t E: JR) •

The normalized polynomial B-splines satisfy the important identity

.Q.

~ i=O

N . (t) n,~

(a < t < b) ,

which is due toMarsden [33].

2.5.2. The condition number of a normalized B-spline basis

As a measure for the quality of a basis serves the condition nu~ber:

broadly speaking, it measures the possible relative change in c as a result

of a relative change in the spline s.

The condition number K(x0 ,x1 , ••• ,xn+.Q.) of the normalized polynomial B-spline

basis N . (i= O,l, •.• ,.Q.), relative to the knot sequence x0 ~ x 1 ~ ... ~ n,l. S xn+.Q. with aS x0 , xn+.Q. ~band x1 < xi+n (i O,l, •.• ,.Q.), is defined by

(cf. De Boor [3])

(2. 5. 2) := ( sup IISII[a b]) 1( inf JISII[a,b]) 1

IIE_I}=l I IIE.IJ=l

T JR.Q.+l IIE.II I where c = (c0 ,c 1 , ••• ,c.Q.) E I ma x

Osis.Q. I and s

Since

.Q.

ls(t) I s IIE.II I N . (t) IIE_II i=O

n,~

because of Marsden's identity 1 we conclude that the numerator in (2.5.2)

equals one. Hence

(2.5.3) \( inf IJ siJ [a

1b])-l •

11,9!=1

46

The condition number Kn of the normalized polynomial B-spline basis of

ordernis then defined as the supremum of K(x0 ,x1, ••• ,xn+t) over all knot

sequences in [a,b], i.e.,

(2.5.4)

x1 < xi+n (i= 0,1, ••• ,t)}

Lower and upper bounds for Kn have been given by De Boor [3] and Lyche [32].

It follows from (2.5.3) that the problem of computing Kn is equivalent to

the problem of minimizing the sup norm of a specific linear combination of

normalized polynomial B-splines over all knot sequences.

With respect to this general problem, which seems difficult to solve, our

objective is modest: instead of trying to solve it, we shall investigate in

this subsectien the simpler problem, formulated by Meinardus [34, p. 174],

of minimizing the sup norm of a single polynomial B-spline of order n.

Using recurrence relation (2.4.15) we are able to give a complete solution.

We note that the exposition given closely fellows a preliminary report of

ours (cf. Ter Morsche [42]) •

2. 5. 3. The computation of the minimum supremum norm

Throughout this subsectien let n € :N be fixed and let x0 ,x1,. j. ,xn be a knot

sequence with 0 x0 s x1 s ••• s xn = 1. Our aim is todetermine

(2.5.5)

where

IIM oll == sup M o<t) • n, tE:lR n,

Before going into details, we give four simple examples of polynomial B­

splines corresponding to specific knot distributions and their sup norms.

These examples serve as an illustration and will be needed later on.

EXAMPLE 1. x1 x2 = ••• =x

n-1 = X € (0 1 1) 1

{ n(~t' (0 $ t $ x) I

M O (t)

n(i = ~r-1 n,

(x < t s 1) I

IIM all n • n,

EXAMPLE 2, x1

XE (0,1), x2

x3

(2.5.6)

M 0 <tl nl

n-1 n-2

I!Mn,oll n(x/ (1- (1 x)n-2))

(0 ~ t < x) ,

(x ~ t < 1) ,

EXAMPLE 3. 0 = x1 ~-1 1 ~ = ~+1 = ••• = xn-1 1 ( 1 s k s n},

M 0 <t> = k(n)tn-k(l- t)k-1 (0 s t s 1) I n, k

(2.5.7) IIM n,o 11 k(n) (n _ k)n-k(k _ 0 k-1

L(n,k) k (n- 1) n-1 =: .

EXAMPLE 4. x. j/n (j 1, ... ,n-1} 1

J

M 0 <tl n,

n r (-1) n-j (~} (nt- j) n-1 j=O J +

(t E JR) 1

(2.5.8) IIM 0

11 = M 0

nJ - _.!._ n, n 1 - 2TT

-i~ n i~ oo n( n) 2

Jn\1-e ~e dw

(iw)n = 2TT

(n + oo) •

47

Computational details concerning the Examples 1 1 2,3 1 4 are omitted here. We

only remark that the formulas in (2.5.8) are obtained by using (2.4.17)

S2 = ... =Sn= 0 1 followed by Fourier inversion; the asymptotic

48

behaviour of the integral in the right-hand side of (2.5.8) follows by an

application of formula (4.2.4) in De Bruijn [8, p. 65].

In order te cernpare the asymptotic behaviour of 11 M 0

11 in Example 4 wi th n,

that of IIM 0

11 in Example 3, we apply Stirling's formula in (2.5. 7). For n, that purpose the following two cases are.considered:

i) if k is fixed, then

(2.5.9) e- (k-1) (k _ 1) k-1

L(n,k)- (k- 1): n

iil if lim k/n = a with 0 < a < 1, then n+oo

(n-+ co)

(n -+ oo) •

Cernparing (2.5.8) and (2.5.10) we note that for sufficiently large n the

sup norm corresponding to equally spaeed knots is larger than the sup norm

corresponding to a distribution of knots as given in Example 3 with a = l in (2.5.10).

Hence, for large values of n, an equally spaeed knot distribution does not

furnish the minimum sup norm.

In what fellows we shall establish that the knot distribution as given in

Example 3 with k U {n + 1)] provides the minimum sup norm over all knot

sequences 0 = x0 $ x1 $ ••• $ x0

_ 1 $ xn = 1. To this end recurrence relatien

(2,4.15) is needed, viz.

(t- x) , n + 1 Mn+1, (x) (t) = --n- Mn+1, (x) (t) + -n- Mn,O (t) (t € ::R) •

This relation may be considered as a differential equatien for the B-spline

Mn+1,{x)' the solution of which is given in the following

LEMMA 2 • 5. 1.

t M 0 ('t')

(n + 1) (x- t) n J

n, d't' (0 $ t < x) (x_ 't') n+l

I

0

(2.5.11) Mn+l, (x) {t) n + 1 M (x) (t = x) n n,O

1 M 0 {'t')

(n+l)(t-x) 0

J n,

dT (x < t < 1) (T- x)n+1

. t

If M 0 is continuous on ~. then differentlation in (2.5.11) followed by n,

49

integration by parts shows that M' 1

( ) can be expressed in terms of M 0 n+ , x n,

as follows

(n + 1) (x- t)n-l t (x -T) -n dMn,O(T)

0

(0 s t < x) ,

(2.5.12) M;+1, (x) {t) M;, 0 <xl (t =x) if Mn+l,(x) is differen­

tiable at x,

n-1 J1

-n (n+l){t-x) (T-x) dM O(T) n, (x < t s 1) •

t

As a first step in obtaining an optimal knot distribution we fix the knots

x1, ••• ,xn-l and study how the value of IIMn+l,(x)ll depends on the position

of the knot x.

the multiplicity of a knot in the interior of [O,l]

obviously does not exceed n-1, hence M is continuous in (0,1). If M 0 n,O n,

is not continuous on R, then either x0 or xn is a knot of multiplicity n.

In this case the dependenee of IIM 1 ( )11 on x is given by formula (2.5.6), n+ , x wi th n replaced by n + 1. Ha ving observed this, we assume from now on that

M 0

is continuous on R. n, *

Let x E (0,1) be the point where M 0

attains its maximum value and let * n n,

xn+l (x) E (0,1) denote*the point where Mn+l, (x) has its maximum. The follow-

ing result shows how xn+l (x) depends on x.

LEMMA 2.5.2.

* then * * If x> x x < xn+l (x) < x , n n

if x< * then * * x x < xn+l (x) < x n , n

if x * then * * x xn+l (x) x n n

* PROOF. If x> xn, then in view of the first formula of (2.5.12) we have

* * M' (x )

n+l, (x) n

xn (x-x*)n-1 (n+l) J ___ n __ dM O(T)

(x_ ··r)n n, 0

> 0 •

50

It can easily be shown by Rolle's theorem that S-(M'+l ( ),(0,1)) = 1 and * * n , x

thus the preceding inequality implies that x <x 1

(x). * n n+ *

Now let x< x. Using again (2.5.12) we conclude that M'+l (}{x)< 0. * *n . n , x n

Hence x >x 1 tx). Taking into account the sign of M'+t () (~.> (cf. n n+ * n ,x; *

(2.5.12)) we likewise deduce that x> x implies x> x*+1

(x), ;x< x implies n n . n * * ' * * ' x< xn+l(x) and, finally, x= xn implies xn+l(x} xn =x. A combination of

these results yields the lemma. D

* * * We preeeed by assuming that x :::: xn' since the two cases x ~ x and x s: x n * n

are very similar. Recalling that Mn+l,(x) attains its maximum at xn+l(x),

we deduce from (2.4.15) that

(2.5.13) n + 1 * IIMn+l, (x)ll .. -n- Mn,O(xn+1 (x))

Since M 0

is decreasing on (x*,l) one has n, n

n + 1 -* inf IIMn+l,(x)ll = -n-Mn,O(xn+l) ,

x* s:xs: 1 n

where

-* We shall prove that (cf. (2.5.16)) xn+1 * xn+1 (l}. For this purpose the

following lemma is needed.

* * * LEMMA 2.5.3. If xn < y < z :S: 1, then xn+l(y) < xn+l(z).

PROOF. From (2.5.12) it fellows that for x > x* n

(2.5.14)

* xn+1 (x)

I 0

-n (T-X) dM O(T) n, 0 •

* Now let xn < y < z s 1. An application of the mean value theerem yields

-n (T- z) dM O(T) n,

0

51

= ( ::..C..:.)n '1-z

+(~)n T 2 - Z

* xn+l (y)

* xn+l (y)

J x

n

{(T - y)n (' - y)n} J

= T:- z - T ~- z * (T - y) -n dMn,O (T) , x

n

forsome values -r 1 E (O,x:l and -r 2 E

AsT+ (~1r is decreasing on (O,y) we have ;: - z}

* xn+1 (y)

(2.5.15) sgn( J (r-z)-ndMn,O(r))=(-1)n.

0

The function h defined by

t

h(t} := J (T-Z}-ndMn,O(T)

0

*

-n {T - y) dM O (T) n,

changes sign on [O,z) only at t xn+1 (z); in fact, we have

sgn(h(t)) ! (-1) n

= (- 1) n+1

* (0 < t < xn+1

(z)) ,

* (xn+l (z) < t < z)

* * so frorn (2.5.15) it follows that xn+l (y) < xn+ 1 (z).

COROLLARY 2 • 5 • 4 •

(2.5.16) 11 11 n + 1 * ( 1 ) ) Mn+1, (x)' = --n-- Mn,O(xn+1

(2.5.17) min IIMn+l, (x) 11 O:<;x:<;x*

n

+ n

* Mn,O(xn+1 (O))

PROOF. Equality {2.5.16) follows irnrnediately frorn Lemma 2.5.3 and (2.5.13).

* Equality (2.5.17) follows frorn the observation that the cases x ~ xn and

x s x* are essentially the same. n

0

0

52

Corollary 2.5.4 implies that the function x._,IIM 1 ( )11 attains its minimum n+ 1 x at x= O.or x= 1.

However, the preceding analysis is carried out under the assumption that

M 0

is continuous on· lR {cf. p.·49), If M 0

is not continuous on. JR, then n, n, either x0 or xn is a knot of multiplicity n. Assume that xn is :a knot of

multiplicity n, then in view of (2.5.6), ·with n replaced by n + 1, one has

n n-1

11Mn+11 (x)ll = (n+l)(x/(1- (1-x)n+1))

From this expression we conclude that in case n ~ 2 the function

X'- IIM 1 ( ) 11 attains its minimum value at x • 1. Evidently1 the assump-n+ , x tion that x 0 is a knot of mul tipHei ty n similarly implies that IIMn+l

1 (x) 11

attains i ts minimum value at x = 0 in case n ~ 2.

The main result in this section is contained in the theorem below.

THEOREM 2.5.5, Let M 0

be the potynomiat B-spUne of order n IA'lth knots n, 0 = x0 s x1 s ••• s xn = 1. Then

(n = 2k) ,

(n "' 2k + ll .

Moreover~ the knot dietn'butions foX' r.1hich the minimum sup no:l'PTI is attained

are given by

xo = x1 = ••• = ~-1 0 , ~ = ~+1 =x = 1 1 n

r.1here r;· (n odà) ~

k = n .!!+1 (n even) • 2 Ol' 2

PROOF. The case n = 1 is trivial. Furthermore1 IIM210 11 = 2 1 independent of

the position of x1• Let now Mn10 be a polynomial B-spline of order n (n ~ 3)

and assume there is a knot xi in the interior of [01lJ. By deleting this . ~1

knot we get the B-spline M(D :x0,x1, •• , 1xi_1,x1+1, ••• 1xn;•). By Corollary

2.5.4 and Lemma 2.5.3 one may conclude that

53

IIM 011 > n,

Thus a knot distribution with knots in the interior of [0,1] cannot be op­

timal and therefore we may restriet ourselves to knot distributions with all

knots located at the end points 0,1. Example 3 on p. 47 furnishes such a

knot distribution and so it can be used to obtain the minimum sup norm.

Consequently, it remains to compute min L(n,k) (cf. (2.5.7)). 1,sk,sn

The theorem will be completely proved if we show that

(2.5.18) n+1 min L(n,k) = L(n,[~])

1<;;k<;;n

This can be done as fellows. In view of (2.5.7) and writing t

have

L(n,k+1) L(n,kl

( )

k-1 1 +k~1

( )

t-1

·1 +t ~ 1

Since the sequence ( ( 1 + 1/k) k) is increasing it fellows that

n- k, we

L(n,k+1) /L(n,k) < 1 if k < Jl., hence k < n/2. This implies (2.5.18). 0

REMARK 1. With respect to the supremum of the sup norm of Mn,O our analysis

shows that

sup IIM 0

11 n • < <1 n,

• • ·-Xn-1-

Indeed, the knot distributions as given by Example 1 have this extremal

property.

REMARK 2. An elementary computation shows that the sequences (p 2k+l}~ and

(p 2k>7 defined by

p2k+l := (2k + 1) -! L(2k+l ,k+l) I := (2k)-~ L(2k,k)

are both decreasing with (cf. (2.5.10))

min o:s:x1:s: ••• :s:xn_ 1:s:1

(2n)! IIM 11 ~ -n,O 'Tl

lim P2k k-

lirn P2k+1 k-

(n-+ oo) •

He nee

54

REMARK 3. It can also be shown that the sequence (pk)~ satisfies

Hence, for n 3, 4, •••

min osx1s ••• sxn-1!!>1

{k = 2,3, ••• ,)

where the constant B/9 = O.BBBB ••• cannot be replaced by a smaller one. On

theether hand, for n = 1,2, •••

where the constant (2/~)~ 0.7978 ••• cannot be replaced by a larger one.

REMARK 4. For an arbitrary knot distribution x0 !!> x1 !!> ••• ~ xn with x0 < xn

the function

is a polynomial B-spline of order n having its first kno~ at 0 and its last

knot at 1. In view of (2.5.19) one has

(n = 1,2, ••• ) •

REMARK 5. As a consequence of (2.5.20) and (2.5.1) one has

IIN 11 :<: (.3..)! n, 0 nrr (n = 1,2, ••• ) ,

where N 0 is the normalized polynomial B-spline of order n as defined by n, (2.5.1).

Hence, (2.5.3) and (2.5.4) imply that K :<: (n~/2)1. However, this lower n

bound of Kn is poor since it has been shown by Lyche [32] that

3

(2.5.21) n-1 n2 Kn :2: -n- 2

We Cbserve that the lower bound in (2.5.21) is obtained by taking the knots

in [ 0, 1] distributed as equally as possible over the endpoints 0 ,1.

3. ON RELATIONS BETWEEN FINITE VIFFERENCES ANV VERIVATIVES OF CARVINAL .C -SPLINE FUNCTTONS

3.1. I~oduction and ~umm~q

55

It is well known (cf. Ahlberg, Nilson and Walsh [1, pp. 10, 12]} that for

a cubic spline, i.e., a polynomial spline s of order four (or degree three)

with equally spaeed knots xi = ih (h > 0, i € ~) the following relations

hold:

{3.1.1)

{

{s" (xi+2l + 4s" (xi+l} + s" (x i})

{s' {xi+2) +4s' {xi+l) +s' (xi))

6 = 2 (s (xi+2) - 2s (xi+1) + s (xi)) , h

3 h {s (xi+2) - s {xi))

For quintic splines, i.e., polynomial splines of degree five, relations

similar to (3.1.1) have been derived by Schurer [57] and for polynomial

splines of arbitrary order by Fyfe [18]. Note that in these relations the

spline function s and its derivatives are evaluated at points that coincide

with the knots. These relations are often used in problems of interpolating

a gi ven function by means of splines in case the points of interpolation and

the knots coincide. From now on the points of interpolation will be called

node a.

In order to obtain results with respect to the problem of existence and

convergence of (periodic) spline interpolation in the case each node is

located between two consecutive knots, relations similar to {3.1.1) are

required. This is done by Ter Morsche [40]. Relations of the kind (3.1.1)

can be written in a compact form using the shift operator E as introduced

in Definition 2.4.5; in the case of polynomial splines we refer to Ter

Morsche [41].

The objective of this chapter is to derive relations similar to {3.1.1) for

cardinal .C-splines with respect to an arbitrary differentlal operator pn(D).

Fundamental for the derivation of these relations, which will be called

relations between finite differences and derivatives of cardinal .C-splines,

56

are the so-called exponential .C -spline functions. In this respect i t is

appropriate to cite Schoenherg's remark about these functions for polynomial

splines (cf. Schoenberg [51, p. V]): "the role played by the exponential func­

tion in calculus is tàken over in equidistant spline theory by the exponen­

tial spline". A similar observation applies to the exponential, .f·-splines in

the case of the, more genera!, .f-spline theory.

The contents of this chapter may be briefly summarized as fellows.

Sectien 3.2 introduces exponential.C-splines and contains various properties

of them. Part of it is preliminary material for Sectien 3.3, in which a few

fundamental relations between finite differences and derivatives are derived.

In the final Section 3.4 a lemma will be given that is of importance for the

problem of cardinal.C-splineinterpolation, tobedealt with in Chapter 4.

Moreover, a number of simple differentiation formulas arededuced.

3. 2. The exponen:t,La,t t--6pUnu

3. 2. 1 • De fini ti on of exponential .C -splines

Throughout this chapter pn € ~n will be a menie polynomial and h a positive

number. Associated with pn and h, the mesh distance of the knot distribu­

tion, we introduce the polynomial Pn as :ollows: if a1,a2, ••• ,an are the,

not necessarily real, zeros of pn, then pn is given by

(3.2.1) n B.h

:= n (z-e J ) j=l

(z € f::) •

Furthermore, we reeall (cf. 1.3.2)) that S(pn,h) denotes the s.et of cardinal

.f-splines corresponding to the operator pn(D) and with mesh distance h.

In order to define the exponential .C-splines we need the following elemen­

tary result, a proof of which may be given by induction with respect to the

variable n.

LEMMA 3.2.1. If À E: JR is suah that pn(>.) '# o, then the nuU function is the

onZy funation f E: Ker(pn) satisfYing

f(t +hl = Àf(t) (t € JR) •

57

Given an arbitrary c # 0 it follows that under the condition of Lemma 3.2.1

there exists a unique f E Ker(pn) such that

(3.2.2) f (i) (h) - H(i) {0) = cö. J.,n-1 (i 0 1 1 1 • • • 1 n-1 ) •

DEFINITION 3.2.2. Let À E lR be suah that pn (À) # o. Then a nontrivial t­

spline s E s (pn,h) is called an exponential t-spline function if it aatis­

fiea the functional equation

(3.2.3) s(t+h) = Às(t) (t E JR) ,

Si nee an exponential t -spline s belongs to c (n- 2) (lR) i t fellows that

(i) (i) . s (h) - Às {0) = 0 (J. = 0 1 1, ... 1 n-2) , and thus by (3. 2. 2) an exponential

t-spline is uniquely determined by (3.2.3) up to a multiplicative constant.

If pn(D) is disconjugate on [Oinh] 1 then an exponential t-spline correspond­

ing to pn(D) can easily be expressed in terros of B-splines. This is the

content of the following

THEOREM 3.2.3. Let the Óperator p (D) be disconjugate on [Oinh] and let n

À E lR be such that p (À) # o. s E S(p 1hl is an exponential t-spline, n n then there exists a constant c # 0 such that

(3.2.4) s(t) = c (t E JR) 1

~here (cf. p. 29 for notation) Bh(pn;•} denotes the B-epline of order n

aarreeponding to the operator pn(D) and having equidistant knots O,h, ... ,nh.

PROOF. Since pn{D) is disconjugate on [ih,ih+nh] for every iE~, we may

apply Lemma 2.2.2 in order to obtain

"" s (t) I cvBh(pn;t-vh}

v .. -oo

(t E JR) 1

where the coefficients cv are uniquely determined.

Since (3.2.3) holds these coefficients satisfy the relation

(V E :iZ) ,

Hence c = cÀv for some constant c # 0. \)

D

58

I f f E Ker {p n) has the property that i t coincides wi th an exponential .C -

spline on [ 0 ,h) , then f is called an eg:ponential t-polynomial. Furthermore,

exponentlal t -polynomials corresponding to p (D) = on are simply called n

ereponential polynomiala.

It fellows from (3.2.2} that an exponentlal t-polynomial satis;fies the

relation

(3.2.5) f(t+h) -Àf(t) =ctp(t),

where (cf. p. 6) ~ is the fundamental function corresponding ~ pn(D) and c

is a constant different from zero.

On the other hand, if p (À) # 0 and if c is prescribed, then there exists n

exactly one function f E Ker(pn) satisfying (3.2.5).

We preeeed by giving an example of a particularf.-spline corresponding to

the operator pn+l (D) "' Dn+l. Given c E lR with c # O, let p E lfn be the po).y­

nomial of degree exactly equal to n satisfying

(3.2.6) p(t +h) + p(t) (t E lR) •

The polynomial p is well known and can be written as

{3.2. 7) p(t)

where En is the classical Euler polynomial of degree n (cf. Nörlund [45,

p. 23]) given by the generating function

2etz ---= ~ < I z I < 2lT) • ncO

The exponentlal t-spline corresponding to Dn+l and having the property that

it coincides with p on [O,h) satisfies (3.2.3) with À = -1. This is a

specific so-called Euler f.-spline, the general definition of which reads as

follows.

DEFINITION 3.2.4. Let pn(-1) # 0, If sE S(pn,h) satis;fies the functional

equation

s(t+h) =- s(t) ( t E ll.) 1

then s ia aaUed an Eule:I' .C-apUne j'unation. In pal'tiaula:zr, if pn (D) "' D0

then s is aalled an Eule:I' spline jUnotion of o:I'de:I' n (o:I' deg:I'ee n-1),

59

3.2.2. The normalized exponential !-spline

We reeall that an exponential !-polynomial is uniquely determined up to a

multiplicative constant. Prescrihing the constant c in (3.2.5) in a particu­

lar way gives rise to the normalized exponential l-polynomial.

DEFINITION 3.2.5. Let À E ~ be suah that (af, (3.2.1)) p (À) ~ 0, The n

normaZized exponentiaZ !-poZynomiaZ ~À (pn;h,•) ia the unique exponentiaZ

!-poZynomiaZ satiafying

(3.2.8) (t E" J<) 1

uhere~ as usuaZ~ ~ ia the fundamental jUnation aorresponding to the operator

pn(D).

The normalized exponential !-polynomial is now used to define the normalized

exponentlal !-spline function in the following way.

DEFINITION 3.2.6. Let À E :R be auah that pn(À) 1 0, The normalized exponen­

tial l-apline funation '!'À (pn;h,•) ia the unique exponentiaZ !-spline satis­

fying the two aonditions

i) (t E J<) 1

ii) (0 "' t < h)

If confusion is unlikely we shall simply use the notations '!'À and tj!À in­

stead of '!'À (pn;h,•) and tj!À (pn;h,•).

3.2.3. Some properties of the function tj!À

In the following lemma a number of properties of the normalized exponential

!-polynomial $À is collected. In accord:nce with Definition 3.2.5 it is

assumed throughout this subsectien that pn{À) 1 0.

LEMMA 3.2.7. The funation $À as defined in Definition 3.2.5 enjoys the

foZZowing properties:

i) $À has the representation

60

(3.2.9) p (À) ~ tz

Ijl;. (t) = ~i r --7~-z--- dz (;.- e ) Pn (z)

(t e :R) ,

c

~here c is any aontoUP in the aomplex plane sUProunding the zeros of . tz hz -1

p but exalud1-ng the po les of z He Ot- e l . n

ii) Por any t e R the funation ;."""""'Ijl;. (t)' is a polynomial of degree at most

n- 1 ~ith leading aoeffiaient Q> {t) •

iii) If pn {Dl is disaonjugate on [O,nhJ, then 'ii;. admits the Pepresentation

n-1 (3.2.10) 1P;.<tl = c lo i.vBh(pn;t+(n-1-v)h) (0 $ t < h) ,

iv)

~here the aonstant c does not depend on i..

Let the polynomial p~ be defined by p: (t) :=

(1.4.24)), then

(-l)n p (-t) (af. n

(3.2.11) * n~* n-1 'ljl,(pn;h,h-t) = (-1) p (O)À ~ 1 {p ;h,t) " n ;.- n

(t e :R) •

v) Por suffiaiently small I Al the funation 1P;. aan be ~ritten in the form

(3.2.12) 1/\(t) =- pn(À) Ï rp(t- (k+1)h)Àk k=O

PROOF.

(t e :R) •

i) Let f, considered as a function of t, be defined by the right-hand

side of (3.2.9). Then f e Ker(pn) since the contour C excludes the poles of hz -1

z >--+ (À - e ) • Moreover, one readily verifies that

It is well known that

1 h etz (3.2.13) rp(t) = 2'd j p (z) dz ,

C n

so we conclude from (3.2.8) and the uniqueness-of Ijl;. that Ijl;.= f, i.e.,

that {3.2.9) holds.

ii) Assume for simplicity that Pn has distinct zeros a1,a2, ••• ,Sn' then

according to the residue theerem property (3.2.9) implies

61

(3.2.14)

In view of (3.2.1) we observe that À~ljJÀ(t) is a polynomial in À of degree

at most n- 1 with leading coefficient

which, by the residue theorem applied to (3.2.13), equals ~(t). Thus proper­

ty ii) is established in case pn has distinct zeros.

If pn has multiple zeros, then ii) may be obtained by considering the dis­

tribution of the zeros at hand as the limit of a sequence of distributions

of distinct zeros.

iiil This property follows from (3.2.5) and the observation that

iv) ~ * In view of (3,2.8) the function ljJÀ(pn;h,•) satisfies the functional

re lation

Moreover, we note that

( -1) n-1 q> ( t) •

Consequently,

and hence (cf. (3.2.8))

(-1) n P* (0) À n- 1 1jJ

1 (p ;h, t)

n À- n

v) . tz hz -1 This property follows from (3.2.9) ~f we replace e (À- e ) by the

series

I (t-(h+1)k)z ,k e A I

k=O

which converges on the. contour C if I À I is sufficiently small. 0

62

Assuming pn(D) = Dn (polynomial case), together with h

(3.2.12} takes the form

"' (3.2.15) n · n \' 1/IÀ(D;1 1t)=(1-À) l.

k=O

(k + 1 - t) n-1 À k

(n- 1) !

1 1 we observe that

Comparing this result with formula (3.1Ó) for P 1 in Ter Mor~che [40,p.212]1 n-we abserve that lj!À (Dn;l1•) agrees with Pn-l up to the multiplicative

constant (n- 1)!. Consequently, properties of normalized expopential poly­

nomials (we reeall that in the polynomial case the t. is omitted from the

notation) can be obtained from results that have been proved for Pn-l in

Ter Morsche [401 p. 212]. In the following subsectien we list some of these

properties.

3.2.4. Some properties of the normalized exponential polynomials

In this subsectien lj!À denotes the normalized exponential polynomial, i.e., n

1jJ À : = Ij! À (D ; h, •) •

According to Ter Morsche [40,p. 212] and taking into account the results of

the previous subsection1 we observe that the function ljJÀ has the following

properties:

{3. 2.16) 1/IÀ (t + h) - ÀljJÀ (t)

(3.2.17)

(À_ 1) n hn-1 tn-1 (n- 1) ! (t E lR) 1

in which Il 1 is the so-called Euler-Frobenius poZynorrrlaZ of degree n- 2 n-(cf. Schoenberg [51 1 p. 22]~ where an interesting survey of the main proper-

ties of these polynomials is given). The first few Euler-Frobenius poly­

nomials are

r··" 1 li3 (À) À 2

+ 4À + 1

{3.2.18) lil (À) 1 li4 (À) À3+tH2 +1D.+1

n2 (À) À+l li5 (À) À 4 + 26À 3

+ 66À 2 + 26À + 1

(3.2.19)

where En-l is the Euler polynomial of degree n- 1 (cf. p. 58) ,

(3.2.20) Cl n

<lt ljJÀ (0 ;h,t) n-1

(À- l)ljiÀ (0 ;h,t)

(3.2.21)

(3.2.22) 1/JÀ (t) tn-1 Ij; -1 (1 - t) À

63

3. 2. 5. Extension of the de fini ti on of the normalized exponential .C-polynomial

We reeall (cf. p. 59) that the function 1/JÀ (pn;h,•) is defined for those

values of À for which p (À) ~ 0 and that (cf. property ii) of Lemma 3.2.7} n

À~ lj!À (pn;h,t) is a polynomial. This gives rise to the following

OEFINITION 3.2.8. Let ~ E t be sueh that p (~) 0. Then the normalized ex-n

ponential .t:-polynomial w ~ (pn;h, ·} is defined by

(3.2.23) lJ! (p ;h,t} := lim l)J, {p ;h,t) a n À+a A n

{t E :R)

In view of (3.2.8} the function 1/Ja satisfies the functional relation

{t E :R)

If there is precisely one zero 6 of pn satisfying e 6h a, then, since

pn(D)1/Ja(t) = 0 (tE :R), there exists a constant ca such that

(3.2.25) (t E :IR) •

The constant ca in (3.2.25} may be obtained by applying the residue theerem

in (3.2.8); we cbserve that we only have to compute the residue at z = 8.

Hence

(3.2.26) c a

1 lim p (À) Res ---:-'----À+a n z=6 (À-

where the polynomial qn-k is given by qn-k(z)

the multiplicity of 8.

-k := (z- 6) pn (z), k being

81h B2h If there exist at least two different zeros 61 and 82. such that e e

~ a, then a short calculation shows that

64

tz lim p (À) Res ---:e ___ _

n hz À-+il z=B (À- e )pn (z)

0 (t € JR) •

- Bh . REMARK. If pn(a) = 0 and a= e < O, then B is a nonreal zero of Pn· Since

p is a polynomial having real coefficients, the complex conjugate ä of B n hä

is a zero of pn which also satisfies e = a. It fellows from the foregoing

that w_1 (pn;h,t) = 0 {t € lR).

3.2.6. on the zeros of the normalized exponential l-polynomials

The first objective of this subsectien is to prove a result similar to the

following

LEMMA 3.2.9 (Michelli [37,p.210]). Let p e ~ be a monia potynomial having n n

only reat zeros. Then for eaah h > 0 and eaah À< 0 the jUnation ~À(pn;h,•)

has eroaatly one zero in [O,h).

If pn has zeros not all of which are real, an additional condition has to

be formulated to ensure that wÀ has precisely one zero in [O,h). This is

specified in the following

LEMMA 3.2.10. If the operator p (D) (n ~ 2) is disaonjugate on [O,(n-l)h]~ n

then for every À < 0 t'he funation wÀ (pn;h,•) has preaisely one zero in [O,h)

or> ~,_Cpn;h,•) is identiaaUy zero.

PROOF. We may assume that ~À is a nontrivial function. Since ~À E Ker(pn)

is an analytic function, each of its zeros is a streng zero in lR having

multiplicity not exceeding n- 1 with respect to any sequence of differentlal n-1 operators (D1 >1 • As the operator Pn(D) is disconjugate on [Q,(n-1lh] we

may·write (cf. (1.4.8}): pn(D) = o[n] = Dn Dn-l •••D1• Taking into account

Corollary 1.4.22 one has

(3.2.27)

If 1PÀ (0) = -lj!À (h) .f 0 then Ij\ has an odd number of zeros in (O,h). Con­

sequently, the assumption that w,_ has at least three zeros in [O,h) I combined

with the functional relation (cf. i), p. 59) 'I' À (t +h)

n-1 Z((Dil 1 ,'I\,(O,(n-1)h));:: 3n-3

À 'I' (t), implies

which contradiets (3.2.27). Hence ~À has precisely one zero in [O,h).

If 1/JÀ (0) -1/JÀ (h) = 0 and if 1/JÀ has at least one zero in (O,h), then o 1~À has at least two zerosin (O,h). Hence,

n-1 Z((Di) 2 ,D1'1'À,(O,(n-1)h)):?: 2n-2

On the other hand, it follows from Corollary 1.4.22 that

65

Again we conclude that ~À has precisely one zero in [O,h). This proves the

lemma. 0

Next 1/JÀ is considered as a function of À. In view of Lemma 3.2.7 (property

Hl) the function À~~À (t) is a polynomial of degree n- 1 in case qJ (t) # 0

and of degree at most n- 2 if <p (t) 0.

With respect to the polynomial case pn(D) Dn, it is shown in Ter Morsche

[41] that the polynomial À;---.1/JÀ (t) has n- 1 distinct negative zeros if

0 < t < h and n- 2 distinct negative zeros if t = 0. Michelli establised

that the same conclusion holds if pn bas only real zeros. In fact, one has

LEMMA 3.2.11 (Michelli [37, p. 211]). Let p E ~ be a monia polynomial n n having only real zeros. Then for all t [O,h) the polynomial :\~1/JÀ(pn;h,t)

has only distinct negative zeros.

If not all zeros of pn are real, there are, to the best of o~r knowledge,

no general results available with respect to this problem. The following

simple example shows what kind of situations may occur.

2 EXAMPLE. Let the operator p3 (o) be given by p 3 (D) = D(D +I). Using (3.2.9)

we obtain

(3.2.28) 2

1/JÀ(t) = (1-cos t)À + (cos(t-h) +cos t-2 cos h)À+l-cos(t-hl .

This formula yields the following conclusions:

66

i) 2 if h = 1! and t = 1l' /2, then lj!À (t) = À + 2À + 1 and thus À -1 is a

negative zero of multiplicity two:

ii) if h = 21T and t = 31T/2, then lj!À (t) = À2 - 2À + 1 and thus À • 1 is a

positive zero of multiplicity two;

iii} if 0 < h < 1l' and t h/2 1 then À ~ lj!À (t) has two distinct negative

zeros.

In fact1 the assumption 0 < h < 1T ensures that À~lj!À (t) has two distinct

negative zeros for all t € (01h).

With respect to the general case we conjecture that the disconjugacy of

pn(D) on [01nh] will be sufficient to prove the assertien in Lemma 3.2.11.

3. 2. 7. The perfect Euler l-splines

We reeall that perfectt-splines are introduced in Definition 1.3.5.

A particular kind of these splines, the so-called perfect Euler t-splines1

play a significant role in Chapters 5 and 6. Their definition reads as

follows.

DEFINITION 3.2.12. Let the mesh distanaeh besuah that pn(-1) 'Î 0. A fitna­

tion s, defined on lR, is said to be a perfeat EuZ.er l-spUne of order n + 1

aarreeponding to the operator pn(D) if it satis~es the conditions

r s €. c<n-ll (lR) 1

(3.2.29) ii) s(t+h)=-s(t) (t € JR) I

iii) pn(D)s(t}=-1 (0 < t < h) .

It follows from Lemma 3.2.1 that the condition p (-1) -# 0 ensures that sis n

uniqualy determined by (3.2.29). A perfect Euler l-spline will be denoted

by f(pn;h~·).

In view of (3.2.9) there exists a nonzero constant c such that for 0 < t < h

where cis any contour in the complex plane surrounding the zerosof zpn(z) . tz hz -1 but excludl.ng the po les of z H e ( 1 + e ) •

Since pn(D}E(pn;h,t} = -1 (0 < t < h} the constantcis given by

-1 c

Hence c - 2 and so

(3.2.30) ïli

- ~ .

dz (0 < t < h) •

We reeall that the introductory section of this chapter contains a number

of relations between derivatives and finite differences of a polynomial

cubic spline. The purpose of this section is to generalize this kind of

relations to cardinal t-splines corresponding to an arbitrary operator

67

p n (D) • The main resul ts are Theorema 3. 3. 2 and 3. 3. 3, the contents of which

for the polynomial case p (D) = Dn may be found in Ter Morsche [41]. . n

DEFINITION 3.3.1. Let the normaZized exponentiaZ !-poZynomiaZ be given by

(cf. property ii), p. 60)

( 3. 3.1) n-1

ljJÀ (pn;h,t) = L v=O

a (t) À v V

Then the operator WE(pn;h,t) is defined by

(3.3.2) n-1

ljJE{pn;h,t) = L v=O

a {t)E\1 , \)

where (of. Definition 2.4.5 and p. 44 for notation) Ef(x) := f(x +hl (x E :R).

We shall now prove the following fundamental result from which the required

relations between derivatives and finite differences may easily be derived.

THEOREM 3.3.2. LetsE S(p ,hl, the cZass of cardinaZ t-splines correspond­n

ing to the operator pn{D) and with mesh distance h. Then

68

(3.3.3)

whe!'e \l € 12: aru1 0 s x ::; h, 0 s y s h.

PROOF. The proof will.be divided into several parts. We first show that

(3.3.3) is valid for functions s € Ker(pn), Then {3.3,3) is shown to held

for the Green's function ~+ corresponding to pn(D), under the additional

condition that Pn has distinct zeros. Having established this,,we point out

that this additional condition can be dropped without difficulty. The last

step then is to prove (3.3.3) for a general cardinal l-spline; this is done

by making use of its representation (1.3.1).

As for the first part of the proef, let s € Ker(p ) • It is then sufficient j Bt . n

to prove (3.3.3) for the functions tHt e (J == O,l, ••• ,k-1) 1 where

B E ~ is a zero of multiplicity k of p • We can even restriet ourselves to n k-1 Bt

a proef of {3.3.3) for the function tHs(t) = t e 1 since (3.3.3) then

holds for t~(t-\lh)k-l efl(t-ph) (lJ € ?Z) and it suffices to cbserve that

tj eBt is a linear combination of these functions and that $E is a linear

operator. Now let e be an arbitrary positive number. We define the poly-

nomial p by n1e

k-1 p {z) := qn-k(z) ·" (z-a- je} ,

n 1 e J==O

-k where (cf. p. 63) qn-k is given by qn-k (z) == (z -13) pn (z).

Moreover 1 let the functions s. (j = l, •.• ,k) be defined by J,€

s. (t) := e(fl-(j-l)e:)t ),e:

{t € lR) •

According to (3.3.2) and (3.2.25)

(3.3.4) $E(p ;h,y)sj (ph +x) n,e: ,e: $ (h)(p ;h,y)s. (ph+x) sj,e: n,e: J,e:

== c (h)sj {ph+x+y) = $E(p ;h 1 x)s. {Jlh+y) • s. ,e: n,e: J,e: J,e:

We now introduce the function se: as the divided difference (cf. Subsectien

1. 4. 8) of the function T...,. e tt wi th respect to. the k points 13 - (k-1) e:,

13- (k-2)e:, ••• ,a. In view of (2.2.7) one has (cf. p. 29 for notation)

tk-1

(k - 1) !

00

f Bk-l,e:(t-fl+(k-lle:)ett d-r

-oo

and so

(3. 3. 5) tk-1

(k - 1) : Bt

e

Sinces is a linear combination of s 1 , ••• ,sk it follows from (3.3.4) e: , e: , e: that

(3. 3 .6) ljJE(p ;h,y)s (llh +x) = ljJE(p ;h,x)s (1Jh +y) . n,e: e: n,e: e:

69

Letting e: + 0 in (3.3.6) and using (3.3.5) one obtains (3.3.3), thus showing

that (3.3.3) holds for all sE Ker(pn).

The next step is to prove (3.3.3) for the Green's function ~+ corresponding

to the operator pn(D). In view of (3.3.1) this implies that we have to es­

tablish

(3.3. 7) n-1 I a)yl <p + ()Jh + vh +x)

v=O

n-1 I av(x)<p+()Jh+vh+y) ,

v=O

where 1.1 E 7Z and 0 :> x :> h, 0 :> y :> h.

We observe that two cases can be dealt with easily: if 1J :> -n then (3.3. 7)

holds trivially; if 1.1 :::: 0 then cp+ may be replaced by cp E Ker(pn), for

which (3.3.7) has already been shown to be true.

Now let 1.1 -kin (3.3.7), with k = 1,2, ••• ,n-1. It then follows from the

above considerations that we have to show that

(3.3.8)

is symmetrie in x and y for k = 1,2, ••• ,n-1.

Assume first that pn has distinct zeros B1,s2, •.. ,Bn (BiE tl. Wedefine the

polynomials qj E nn-l (j 1, ••. ,n) by

(3. 3. 9) -1

q.(z) := (z-13.) pn(z) J J

and thus ( cf. ( 3. 2. 1) )

q.(zl = n (z-/i\ . J i;olj

It follows from (3.2.9) and the residue theerem that ljJE(pn;h,y) can be

written as

n 13.y t e J ~ l.. 'P'fiï} qj (E)

j=1 n j

70

Hence

(3.3.11)

~ •. being the coefficient of z1

in q.(z) • .. , J J '

Another application of the residue theerem yields that ~. . can be repre­.. ,J

sented in the form

(3.3.12) 1 L pn(z)

l:.t,j = 21fi j --..:;ll:..jh--!1.-+-1 dz , c {z-e ) z

where C is a closed contour surrounding z = 0.

Substituting (3,3.12) into (3.3.11) we obtain (cf. {3.2.13))

Now let the function H be defined by

(3.3.14) H (x) n-1 : = I cp (x + ( .t-k) h) .t=k /,+1

(x E. :R) •

Then pn(D)H(x) = 0 for x E. :Rand

H(x+h) - ZH(x) cp (x + (n-k) h) cp (x)

--k-z

On account of this and (3.2.8) one bas

(3.3.15) H{x) {Ijl z {p n; h, x) _1/!;;;.z _<P..;;.n:..;_h_, x_+_<_n_-k_l_h_) }

pn (z) zk - zn

-Denoting by C a closed contour surrounding all zeros of pn(z) and z 0, we

cbserve that

(3.3.16) Q I

since the numerator of the integrand is, with respect to z, a polynomial of

degree at most 2n-2 (cf. Lemma 3.2.7, property ii)), whereas 'the denomina­

tor has degree 2n.

Finally, substituting (3.3.15) into {3.3.13) and using (3.3.16) we have

71

f 1/! tpn;h,y)lj! (p ;h,x)

z z n d k ~ Z 1

C z pn(z)

which clearly is symmetrie with respect to x and y. Consequently, (3.3.3)

holds for the Green's function ~+ ûnder the additional assumption that the

zeros of pn are distinct.

If pn has multiple zeros then. formula (3.3.17) may be obtained by a

limiting procedure; we omit the details.

In order to prove (3.3.3) for a general cardinal t-spline s, we note (cf.

(1.3.1)) that intheinterval [)Jh,()l+n)h] scan berepresentedas a linear

combination of a function g E Ker (pn) and the functions t >-+ ~ + (t- jh)

(j = )J+l, •.. ,)J+n-1). Since all functions in that representation satisfy

(3.3.3) we conclude that (3.3.3) holds for any cardinal !-spline s E S(pn,h).

This completely proves the theorem. 0

REMARK. The proof of Theerem 3.3.2 can be greatly simplified if we assume

that pn(D) is disconjugate on [O,nh]. In that case the B-spline representa­

tions (3.2.10) and (2.1.13) are available. The validity of (3.3.3} canthen

be shown as fellows:

"' c L Bh(pn;y+(n-1-v)h)s()lh+vh+x)

\)::::::-00

"' "" = c I Bh(pn;y+(n-1-v)h) I ei Bh (pn;x+()l+V-i)h)

\l=-oo i=-~»

00 00

= c ! c. ! Bh(pn;x+(n-1-a)h)Bh(pn;y+(a+)l-i)h) ~

i=-"' a=-"'

"' 00

c I Bh(pn;x+(n-1-cr)h) I ei Bh (pn;y+(cr+)J-i)h) o=-co i=-"'

= c I Bh(pn;x+(n-1-cr}h)s(\lh+crh+y) = 1/JE(pn;h,x)s()Jh+y) a=-oo

where in the course of writing down these identities we have replaced v by

a := n- 1 - )l- v +i. Changing the order of summatien is permitted, since all

but a finite number of elements àre zero.

72

The follow+ng theorem contains the required relations between derivatives

and finite differences of cardinal l-splines.

THEOREM 3.3.3. Lets E S(pn,h) and Zet p0

be of the /01'171 p0

= p0_kpk, wher>e

Pn-k E 1Tn-k and Pk E 1Tk a:r>e both manie polynomiaZe. Then

-(3.3.18) 1f/E(p0

;h,y)pk(D)s(JJh+x) == 1f/E(pn-k;h,x)pk(E}s(1Jh+y) ,

wher>e JJ E 7l and o s x s h, o s y s h.

PROOF. We note that pk(D)lf/À(pn;h,•} is an exponential l-polynomial eerre­

sponding to the operator p0

_k(D) and satisfying the functional relation

(cf. (3.2.8))

pk(D}1f/À(p0

;h,x+k) - Àpk(D)WÀ(pn;h,x) ==- p0 (À)pk{D}~(x) •

Since pk(D)~ is the fundamental function corresponding to the operator

Pn-k(D} it fellows from (3.2.8) that

(x E :R} •

By taking the "pn-k (D}-derivative" of both sides of (3. 3.3} with respect to

x and using (3.3.19), relation (3.3.18) immediately follows. 0

As an application of (3.3.18) we give a relation between second derivatives

and second order differences of quintic splines; this relation also occurs

in Schurer [57], where it is derived by different means.

For that purpose the Euler-Frobenius polynomials TI5 and IT3 as igiven by

(3.2.18) are needed. In view of (3.2.17} one has

(3.3.20) 4 h3

wÀ co ;h,Ol == 3T n3 (À) , 6 h

5 wÀ co ;h,Ol = 5 ! n5 (À) •

Substituting (3.3.20) into (3.3.18} with x= y == 0 we obtain, for any 1J E ~.

s" ( (1.1+4) hl + 26s" ( (11+3) h) + 66s" ( (ll+2) h) + 26s" ( (ll+1) h}; + s" (llhl

20 N - N = 2 {p2 (E)s((1J+2)h) +4p2 (E)s((IJ+1}h) +p2 (E)s{ph)}

h

20 = 2 {s((p+4)h) +2s((1J+3)h) -6s((1J+2)h) +2s((U+l)h) +s(!Jh)}

h

For 1J 0 this relation does indeed agree with formula (12) in Schurer [57].

73

3.4. Same apptl~ation& on Theo4em~ 3.3.2 and 3.3.3

Chapter 4 deals with the problem of cardinal !-spline interpolation, i.e.,

the problem of determining an s E S(pn,h) with prescribed function values

at the nodes 1-lh +a. ( 1-l Cll), where a. E (0 ,h] is fixed. Replacing y by a. in

(3.3.3),we may interprete relation (3.3.3) as a difference equation for the

unknown s, the characteristic polynomial of which is given by z~--.-1jlz(pn;h,o.).

Being interested in functions s E S(pn 1h) that are of power growth, i.e.,

ls(t) I~ 0(1tlyl (ltl ~ "") forsome y ~ 0, the following lemma will be

needed in Chapter 4.

LEMMA 3.4.1. Let qn E ~n be a monia polynomial that has no zeros on the

unit airale in the complex plane. Furthermore, let the difference equation

(3.4.1) q (E)x b,. n 1-l ,..

(1-l E ;z)

be given, where Exl-l ~~ x]l+l"

the sequence (b1-ll_ .. has the property that forsome nonnegative y

(3.4.2) (11-ll ~ "") f

then (3.4.1} has a unique solution (xl-ll:oo with the property

(3.4.3} lx I 1-l

Moreover, this salution aan be expreseed in the form

(3.4.4) (]l E: 72:) I

where the aomplexnumbere wj are the coeffieiente in the Laurent expaneion

(3.4.5) -1 q (z)

n

that converges on a ring aontaining the unit airale in ite interior.

PROOF. By (3.4.5) and the conditions of the lemma the series E~ lw.l ljly J=-00 J

converges. It is now easy to verify that (3.4.4) is a salution of (3.4.1)

with the asserted property (3.4.3). It remains to prove the unicity of the

solution. Taking into account the general salution of the homogeneaus equa­

tion q (E}x = 0 (]l E ~), we conclude that in this case only the trivial n 1-l

74

solution x = 0 (U € z) satisfies (3.4.3). This proves the unicity of the ).l

solution;

Lemma 3.4.1 mayalso be applied to relation (3.3.18) to the effect that

pk (D) s {x) is expressed in terms of function values of the form s ().lh +y)

D

(1.1 E 10. Such an expression is called a áifferentiation formula. By way of

illustration we give two simple examples concerning cubic splines and quad­

ratic splines.

EXAMPLE 1. The first relation in (3,1.1) may be written in the form

(3.4.6) 3 2 TI

3(E)s' (Uh) = h (E -I)s(llh) (1.1 € iZ) •

As (cf. (3.2.18))

(z + 2 + /3)(z + 2- /:3) ,

n3 has no zeros on the unit circle and thus by putting x11

:= s' (llhl, 3 2 bll := h (E - I)s(l.lh) and applying (3.4.1) one obtains

(3.4.7) s' (0) = ~ Y w. (s (jh + 2h) - s (jh)) j=-co J

In this formula s is the cardinal cubic spline with knots ]Jh (ll € iZ) and

satisfying s(ph) = O<lulyl <lul~ co) forsome nonnegative constant y. It is

easy to verify that the coefficients wj in (3.4.7) are, according to (3.4.5),

explicitly given by

{

(-1) j+1 (2/3) -1 (2 + /3) -j-1

wj = (-1) j+l (2/3) -1 (2- /3) -j-1

(j <!: 0) ,

(j < 0)

EXAMPLE 2. Let p 3 (D) = o3, n = 3, k = 1, x= ~hand y = 0. Relation (3.3.18)

then takes the form

(3.4.8) 1 (E+I)s'(~h+).lh) =h (E+I)(E-I)s(Uh),

As (cf. (3.2.18)) n2 (-1) = 0, we cannot apply Lemma 3.4.1. However, in this 1 particular case i t is easy to see that s' (~hl = h (s (h) - s (0) ) holds for

every cardinal quadratic spline with knots uh (IJ € iZ).

4. ON EXISTENCE ANV CONVERGENCE PROPERTIES OF 1NTERPOLAT1NG CARVTNAL t-SPLINES ANV INTERPOLATING PERIOVIC CARVINAL t-SPLINES

4 • 1 . I nbw du.c:ti.o n. a.nd -6 wnma.Jty

75

Let an arbitrary sequence of real numbers (y~):." be given. We consider the

following interpolation problem.

Determine a function s in S(pn,h), the set of cardinal

sponding to pn(D) and with mesh distance h, that satisfies

(4.1.1) s(~h +a) y~ (~ E ZO) ,

where c; (O,h] is a prescribed nurnber. 00

corre-

If the sequence (y) has oeriod N, i.e., if y N ~ Y" (~ ~ ~) and if ~ _." - ~+ ..

sE S(pn,h) is uniquely determined by (4.1.1), then evidentlysis a peri-

adie function with period Nh. For reasans of brevity, a periadie function

with period T will occasionally be called a T-periodic function and, simi­

larly, a periadie sequence with period N will be called an N-periodic

sequence. Obviously, the problem of periadie t-spline interpolation is

related in a natural way to the more general problem of cardinal

interpolation (4.1.1). The subject of cardinal polynomial spline interpola-

tion in the cases h 1, c; 1 and h 1, a = ! is studied in detail by

Schoenberg [Sl]in his monograph (see in particular lecture 4).

Meir and Sharma [35] investigated the problem of periadie cubic spline

interpolation with variabie a E (O,h] and proved the existence and unique­

ness of a periadie interpolating cubic spline for 0 < a s 1/3 h and for

2/3 h s a s h.

A few years ago we also dealt (cf. Ter Morsche [40, p. 188]) with the

periadie version of (4.1.1) inthepolynomial case and gave a complete solu­

tion of the existence problem. The main result of Ter Morsche [40] is stated

in the following theorem.

76

THEOREM 4.1.1. Let an N-periodia aequenae of real numbera (y~l:® be given

and let a E (0,1] be an arbitrary presaribed number. Then there exieta a

u:nique Nh-periodia spline s E s (Dn, 1) eatisfying the interpolation property

s()Jh+ex) = yll (Jl E E) in eaah of the foll<YWing aaaea:

i) N is odd, n E~ arbitrary, ex arbitrary.

ii) N is even, n is even, ex ,; ~.

iii} N is even, n is odd, ex ,; 1.

The contents of this chapter may be briefly summarized as fellows. In Sec­

tien 4.2 results are derived concerning existence and unicity for the

cardinal L-spline interpolation problem (4.1.1), We note that specific

cases of this general problem have been dealt with earlier; for polynomial

spline interpolation this was done in Ter Morsche [41], whereas Michelli [ 37]

investigated (4.1.1) polynomials p having only real zeros. Sectien 4.3 is n

devoted to the problem of existence and unicity of interpolating peroiodie cardinal !-splines. There an analogue of Theerem 4.1.1 is given for periadie

cardinal !-splines corresponding to operators p (D) with the property * n

pn{D) = pn(D}. The last two sections, Sectiens 4.4 and 4.5, deal with error

estimates for cardinal !-spline interpolation and periadie cardinal t-spline

interpolation, respectively.

Given an arbitrary sequence of real numbers (y )= and an arbitrary positive ~ -=

number h, one may ask under what conditions a function s E S(pn,h) exists

satisfying {4.1.1). As the following lemma shows, the existende of such an

interpolant is always guaranteed if the operator pn(D) is disconjugate on

[O,h].

LEMMA 4.2.1. Let pn(D) (n ~ 2) be dieconjugate on [O,h], Then for every

sequenae (y~l:". and for every ex E (O,h] thesetof functions sE S(pn,h)

satiafying (4.1.1) is a linear ma:nifold of dimeneion n-1 if ex E (O,h),

a:nd of dimeneion n- 2 if ex = h.

PROOF. We first assume that ex E (O,h). It is well known that thesetof

functions f E Ker(pn) satisfying f(cx) = y0 is a linear manifold of dimen­

sion n - 1 • Let f be such a function. In order to construct a function

77

s E s (pn,h) that coincides with f on [O,h] and for which s (\lh +a) ylJ

(IJ E 2l) one may praeeed as fellows. On [h,2h] we put s(t) = f(t) +c1q>(t-h)

(cf. 1.3.1), where the coefficient c 1 is uniquely determined by the require­

ment s (h +a) = y1

, which can be met since q> (a) f. 0. On [-h,O] one has

s(t) f(t) + c_1<p(t). As rp(a-h) 10, the number c_

1 can be ohosen such

that s(- h +a) = y_1

• This process can be continued to yield a function

s E S (pn' hl satisf;ying s (Jlh +a) y \1 (Jl E ll) , which is uniquely determined

by the function f. Hence, for 0 <a < h, the set of functions s E S(pn,h)

satisfying (4.1.1) is a linear manifold of dimension n- 1. If a = h, the

set of functions f E Ker(pn) for which f(O) = y_1

and f(h) y0

is a linear

manifold of dimension n-2. This, in fact, is a consequence of Lemma 1.4.17.

Therefore the set of functions s E S(pn,h) satisfying s(].lh) = y)l-l (Jl E ll)

also is a linear manifold of dimension n- 2. IJ

In case n 1 it is easy to see that the interpolating l-spline s E S(p1 ,hl

is uniquely determined for all a E (O,h], since in this case s is by defini­

tion continuous to the right.

As a consequence of the previous lemma uni ei ty cannot be expected for the

interpolation problem (4.1.1), unless further restrictions on the inter­

polating .C-spline are imposed. Following Schoenberg [51] and Hichelli [37]

we restriet ourselves to sequences {y )00

of power growth, i.e., )1 _ ..

!Y I= 0(1\llyl (IIJI ~ oo) forsome nonnegative constant y. The interpolant s \1

is then required to satisfy ls(t) I O<ltlyl (ltl ~ oo). In conneetion with

this we have the following result.

THEOREM 4.2.2. Let pn

more, Zet (y >"" be a \1 -oo

a E (O,h].

(n ~ 2) be a monio polynomial and let y ~ o. Further­

sequence satisfying IY I= 0(1\.llyl (1\11 ~ <») and let \1

If (cf. Definition 3.2.5) z___, ljlz (pn;h,a) has no zeros on the unit oircZe,

then there exists a unique s E S(pn,h) satisfying the conditions

! . )

(4.2.1) l_

ii)

s (].lh +a) = yll

s(t) = 0( ltiY)

(].! E ]Z) t

(lt] ~ oo)

If ljlz (pn;h,a) has zeros on the unit airale then a function s

satisfying i) and ii) is either nonexistent or nonunique.

78

PROOF. Assume that Wz(pn;h,a) ~ 0 on lzl = 1. First we construct a so-­

called fUndamental solution for the interpolation pxoblem, i.e., a function

L satisfying

(4.2.2) [

i)

ii)

L(J.lh +a) = ö 0 )J,

(}.l 0: ~) I

I L ( t) I decays exponentially if I tI

Since Wz(pn;h,a) has no zeros on the unit circle, we may define numbers

cv o: t (v € ~) by means of the Laurent expansion

(4.2.3) V=-<»

that converges on a ring in the complex plane containing the unit circle in

its interior. Consequently, a number r o: (0,1) exists such that

(4.2.4) Clvl -+ oo)

We reeall (cf. Lemma 3.2.7, property ii)) that

(x o: JR; z o: 4:> •

Moreover, as a consequence of Formula (3.2.9) and taking into account the

secend part of the proof of Lemma 3.2.7, ~ o: Ker(pn) (k = O,l, ••• ,n-1).

Hence we may write for z in a ring containing the unit circle in its in­

terior

(4.2.5)

with

(4.2.6)

wz (pn;h,x)

$z(pn;h,o.)

n-1

co

t c -k ak(x) k=O V

(x E JR) •

( ) ( . ) Since w (p ;h,h) = z lP J (p ;h,O) (j z n z n

O,l, ••• ,n-2) it fellows from

(4.2.5) that

L(j) (h) = L(j) (0) v v-1

(j = O,t, ••• ,n-2) •

In view of this and (4.2.6) the function L defined by

(4.2. 7) L(t) :.: L (t- Vh) -v

(vh < t :;; vh+h ; v o: lZ) ,

79

is a cardinal l-spline: L E S(pn,hl. Furthermore, (4.2.5) implies that

L()Jh +a) = L (a) ~ ö 0

, i.e., -)J \l,

L satisfies i) of (4.2.2). Because of

(4.2.4) and (4.2.6) one has

0 llll L(Jlh+x) = (r l ( I lll + co) •

HenceL also satisfies·ii) of (4.2.2) and therefore L is the fundamental

solution. Now, if (y )00

satisfies y = 0(1\llyl (liJ I+ oo), then the func-\l -"" )J

tion s defined by

(4.2 .8) s (t) := I y\JL(t- Vh) V=-oo

( t E" JR)

belongs to S{pn,h) and satisfies (4.2.1).

In order to prove the unicity of s we note that a function sE S (pn,h) having

the required interpolation property satisfies the relation (cf. (3.3.3) and

(3.3.1))

n-1 n-1 (4.2.9) I

v=O (a)s()Jh +vh +x) L a (x)y

V=O . V ].l+\J 0 ~ x ~ hl •

Since (En- 1 a (x)y )00

is a sequence of power growth, Lemma 3.4.1 ap-V=O v )J+\J \l=-""

plies. This yields the uni ei ty of s.

Now assume that Wz0

Cpn;h,a) = 0 for z0 E ~ with lz0 1 1. We cbserve that

the second part of the theorem will be proved if one shows the existence

of a nontrivial bounded function s in S(pn 1 h) for which s()Jh +a) 0

(\l E ~). Consider the function Wz (p ;h,•) and assurne that Wz (p ;h,•) is 0 n 0 n

a nontrivial function. Then the complex-valued function ~z (p ;h,•) which 0 n

coincides with ~z (p ;h,•) on [O,h] and for which relation i) of Definition 0 n

3.2.6 holds (with À replaced by z0) is bounded and such that

~z0 <pn;h,)Jh+a) 0 (\l E ~). Consequently, the real partor the imaginary

part of o/z (p ;h,•) is a nontrivial bounded function in S(pn,h) vanishing 0 n

at the points )Jh +a (\1 E ~). If tPz (p ;h, •) is identically zero then (cf. ~ 0 n

Subsectien 3.2.5) p (z0) = 0 and there exist at least two different zeros n hS hS s1 and s2 of pn such that e 1 e 2 z

0. Since lz

0! ~ 1 it is then easy

to find a nontrivial g E Ker(pn) having the property g(Jlh +a) = 0 (IJ E 1Z).

This completely proves the theorern.

As a simple illustration we compute the fundamental salution in the case

of trigonometrie spline interpolation of order three (cf. p. 65). The

D

80

oorrasponding differential.J operator is given by p 3 (D) .. O(o2 +I) and the

normalized exponential l-polynomial by (3.2.28). We consider interpolation

at the midpoints of the mesh intervals, i.e., we take a .. !h. A simple com­

putation yields

ljJz(p3 ;h,~h) = (1-cos(!hl)(z2 +2(2 cos<!hl +1)z+1).

For h E (0,~) it is easily verified (cf. the example on p. 65} that

z~ ljJz (p3;h, !hl has two negative zeros a 1 and a 2 satisfying a 2 < -1 < a 1 < 0.

Consequently, Theerem 4.2.2 applies.Partial fraction expansion yields

(4.2.10)

i.e., the Laurent series that converges on the unit circle.

In accordance with its definition (cf. p. 78) the fundamental solution of

the interpolation problem at hand is the bounded function L E S(pn,h)

satisfying L(\.lh +a) = 5 0

(\.1 E lZ). By means of (4.2.5) 1 {4.2.6) and \.1,

{4.2.7) L may be computed on [vh,vh+h] and we obtain

( 4. 2. 11) L ( vh + x)

-1 (al - a2) 1

{ ( 1 ) v+ ( { ) 2 hl v -:(-:-1-_-0-

0-s----.lr-.:-h..,.-) - cos x a 1 + cos x + cos x - h - cos a 1 +

v-1 + (1- cos(x-h))a1 } ,

where v E ~ and 0 ~ x ~ h.

We reeall {cf, Lemma 3.2.11) that the polynomial z~-+ljlz(pn;h,a) has dis­

tinct, negative zeros if p has only real zeros and a E (O,h}. It fellows n . from (3.2.11) that fora= h the polynomial zt-t ljJz{pn;h,h) is zero at 0

while all its other zeros are negative. As a consequence, using Theerem

4.2.2,we obtain the following result, which has also been proved by

Michelli [37] by a different method.

COROLLARY 4.2.3. Let pn e: ~n be a monic poZynomiaZ having onZy T'eaZ zeroa,

and Zet y be a nonnegative constant. CoT'T'eaportding to eve'l'Y aequence of

T'eaZ numbePa (y )00

aatiafYing y = O<lvly> fl\.11 ~ ~J thePe exiata a unique \.1 -~ \.1

s e: S(pn,h) with the p1'ope1'tiea

i) (\l E lZ) ,

81

ii) s (t) (I ti + "') I

if and only if ljJ _ 1 (pn;h,ct) # 0.

In fact, Corollary 4.2.3 states that the interpolation problem {4.1.1) has

a unique salution sE S{pn,h) satisfying the growth property ii) of {4.2.1)

if and only if the Euler .C-spline 'l'_ 1 (pn;h,•) does not vanish at ct.

If pn has zeros not all of whieh are real, general information about the

loeation of the zeros of zH1jl2

(pn;h,a) is laeking, and, consequently,

corollaries to Theerem 4.2.2 similar to Corollary 4.2.3 are diffieult to

obtain.

We finally remark that Jakimovski and Russell [20] have derived some general

results on the existence and unicity of polynomial splines that interpolate

a sequence (y )00

at nonequidistant points; for details the reader is re-IJ -"'

ferred to their paper.

The object of this section is to investigate existence and uniqueness

properties of interpolating periadie eardinal .C-splines.

DEFINITION 4.3.1. A function s defined on the interval [a,b] is said to be

a periadie cardinaZ t-spZine function of order n if s is an .C-spZine with

equally spaeed knots xi = a+i(b-a)/N (i O,l, ••• ,N; NE JN)J and if s

satisfies the periodioity conditions

(4.3.1) s (j) (a) = s {j) (b) (j O,l, •.• ,n-2) •

It is well known (ef. Ahlberg, Nilson and Walsh [1, p. 165]) that for a

given periadie function there exists a unique periadie polynomial spline

of even order (odd degree) that interpolates f at the points xi (i Q 1 1, •

•. ,N); in this case the knots and the nodes coincide. This, in general, is

not true if the periodic splines are of odd order (even degree), i.e.,

interpolating periadie splines of odd order with the knots coineiding with

the nodes may not exist. However, Subbotin [61] proved that an arbitrary

periadie funetion ean be interpolated by a unique per,iodic eardinal poly­

nomial spline of odd order, if the nodes are at the midpoints of the mesh

82

intervals [x. 1,x.] (i = 1,2, ••• ,N). ~- ~

If we take a = 0 and h = {b-a) /N, the periodic continuatien of s to lR is

simply an Nh-periodic cardinal .t:-spline. We shall denote the periodic

cardinal .t: -splines in S {p n ,h) wi th period Nh by

(4.3.2)

Since S{pn,h,N) is a set of bounded functions in S(pn,h), the following

result is a consequence of Theorem 4.2.2.

LEMMA 4.3.2. Let p (n ~ 2) be a monic polynomial and let a E (O,h] be n

such that the polynomial zt--? l)lz{pn;h,a) has nç zeros on the unit airale.

Then for every N-pel'iodia sequenae (y l"" thex>e e:r:ists a unique ]J -«>

sE S(p ,h,N) satisfying s(]Jh+a) = y (J.J €ZZ:). n J.l

PROOF. According to Theoram 4.2.2 thare axists a uniqua bounded function

sE S(pn,hl satisfying s(llh+a) yll. Sinca t~s(t+Nh) also satisfies

this interpolation proparty, it follows that s (t + Nh) = s (t) (t € lR) and

thus sE S(pn,h,N). 0

According to Theoram 4.2.2 tha condition that l)lz{pn;h,a) has no zeros on

tha unit circle is necessary and sufficient to ansure existence and unicity

of a bounded .t:-spline interpolating a bounded sequence of numbers. For

periodic cardinal !-spline interpolation this condition is sufficient;

however, it is not necessary. In fact, the next theorem states that the

condition that z~lj!z(pn;h,a) has no zeros on the unit circle canthen be

replaced by the weaker condition that z~7 l)lz(pn;h,a) has no zeros at the

N-th roots of unity, i.e., lj!z(pn;h,a) F 0 if zN = 1.

Two proofs will be given. The first one is rather long and is constructive

in character. The second one is short and simple, but noneons truc ti ve. Elements

of the longer proof will be needed later; this is one of the main reasons

why it is included here.

THEOREM 4.3.3. Let p (n ~ 2) be a monic polynomial and let N c:. E. Further­n

more, let (y >"" be an. N-periodic eequence and let a " (O,h]. j.l -""

If zHl/lz(pn;h,a) has no zerooa at the N-th roots of unity, then there

e:r:iete a unique s " S(p ,h,Nl eatief.Ying. the interpatation condition n

s {llh +a) = y ll (ll c:. lZ).

83

If z._."lj!2

(pn;h,al has a ze:r>o at an N-th :r>oot of unity, then a funotion

s E S(pn1h1Nl satisfying the inter,polation condition is either nonexistent

01' nonunique.

PROOF.

I. Assume first that $2

(pn;h,a) ~ 0 if zN = 1. Our aim is to construct a

function sE S(pn 1h,N) satisfying the given interpolation condition and,

then, to prove that s is uniquely determined. Let (cf. (3.3.1))

!v (x) := lj!E(p ;h,x)s()Jh+a) I

(4.3.3) \1 n w)J(x): s(x+)Jh) (\1 = 0,1, ••• 1N-1)

An application of (3.3.3) for )J 0,1, •.• ,N-1, together with the pericnicity

conditions for the function s, leads to a system of linear equations Aw = v

of the form

ao(a) aN-1 (a) s (x)

~-1 (a) a0

(a) aN-2 (a) s(x +hl (4. 3.4)

a1

(a) ao (a) s (x+ (N-1)h) VN-1 (x)

where the ak(a) (k = 0 111 •.• ,N-1) are given by (3.3.1) and ak(a) 0 if

k ?> n.

In order to solve (4.3.4) we note that the matrix A in (4.3.4) is a cir­

culant matrix, which we denote by C(a0 (a),a1 (a), •.. ,aN-l (a)). Hence (cf.

Aitken [2, p. 124]) A can be written in the form

(4. 3. 5) A N-1

a0 (a)I + a 1 (a)Q + •.. + aN_ 1 (a)Q ,

where Q is the NXN circulant matrix C(0 11,0, ••• ,0).

We further need the following well-known factabout circulant matrices (cf. 27rik/N

Aitken [2, p. 124]): C(0,1,0, ... ,0) bas eigenvalues wk := e 1 ï

(k O,l, ••• ,N-1) with associated independent eigenveetors (l,wk, ..• ,w~- l

(k O,l, .•• ,N-1). Consequently 1 in view of (4.3.5) the eigenvalues pk of N-1 j

A are pk l:j=O ~(a)wk. Hence (cf. (3.3.1))

(4.3.6) pk = ljJ (p ;h,a) wk n

(k = 0, 1, •.• I N-1 N 2: n) •

84

First let N;:: n. As w: = 1 it follows from the assumption on ljlz(pn;h,a)

that pk # 0 (k = O,l, •.• ,N-1), and thus A is nonsingular. Hence (cf. (4.3.4))

w = A- 1v.

As the inverse of a circulant matrix is again a circulant matrix (cf.

Aitken [2, p. 124]), A-1

has the form C(b0

,b1, ••• ,bN_ 1>. Consequently, for

~ = O,l, ••• ,N-1 and 0 <x s h one has

N-1 (4.3. 7) s (!lh +x) L b . !J!E(p ;h,x)s(jh+a) ,

j=O ~. J n

where (b 0 , ••• ,b N 1J denotes the (~+1)-st row of A- 1 . ' ~. ll, -

Substituting s(jh+a) y. into the right-hand side of (4.3.7) we obtain a J

function s defined on (O,Nh]. Now sis extended to ~ by the relation

s{t +Nh) s(t) (t E lR). The next step is to prove that s E S(pn,h,N) and

that s has the interpolation property s(!lh+a) = y~ (ll = 0,1, ••• ,N-1).

First we note that on each interval (~h,(~+l)h) (!JE~) s coincides with a

function in Ker(pn). Furthermore,

s (k) (~h +) - s (k) (llh- )

N-1 Ï {b . IPE(k)(pn;h,O)s(jh+a)- b IPE(k)(pn;h,h)s(jh+a)}

llt) !l-1,)' j=O

N-1 L b IPE(k)(pn;h,O)s(jh+a) +

. !J-l,j-1 J=O

+ b 0

IPE(k} (p ;h,O) s (a) - b 1 1

lj!E(k) (pn;h;O) s (Nh +a) = 0 \.1, n ll- ,N-

(k = 0, 1, ... , n- 2 ; ~ = 1 , ••• , N-1) •

Hence s(k) (~h+) = s(k) (llh-) (k = O,l, ••• ,n-2) and so h,2h, ••• , (N-l)h are

simple knots of s. In a similar way we can show that s {k) (0 + ) = s (k) (Nh- )

{k = O,l, ... ,n-2). Consequently, sE S(pn,h,N).

We proceed with the proof of s (~h +a) = y ~ (U € Z:) • According to (4. 3. 7) we

must show tha t

This assertion, however, follows immediately from the fact that

C(b0 ,b 1 , ••• ,bN_1

l is the inverse of A. This completes the proof of the

existence part of Theorem 4.3.3 incaseN ~ n. Taking into account the

construction of s we conclude that s is the unique function in S(pn,h,N)

having the desired interpolation properties.

85

Now let n > N and let k be an integer such that kN ~ n. By the proof given

above there is a unique function sE S(pn 1 h,kN) such that s(llh+ct) yll

(IJ E 16). Since t1----+ s (t + Nh) also has these properties it follows that

s(t) s(t+Nh) (tE lR). Hences" S(pn,h,N). This completes the proof of

the first part of the theorem.

Now let us assume that Wz (p ;h,a) 0 n

N 0 for z0 E Ç with z

0 = 1. We note that

the second part of the theorem will be proved if one shows the existence of

a nontrivial periadie function in S(pn,h,N) having the property s(\.;h+a) =0.

This can be done in a similar way as in the proof of Theerem 4.2.2.

II. The restrietion of an !-spline sE S(pn,h) to the interval [O,Nh] can

be represented as (cf. (1.3.1))

N-1 s(t) g(t) + L w.<P+(t-jh)

j=1 J

for an appropriate gE Ker(p) and appropriate coefficients w .• n J

Since dim (Ker (pn)) n, s is determined on [0 ,Nh] by n + N- 1 parameters.

The interpolation condition s(Jlh+a) = yll (Jl = O,l, ••. ,N-1) yields N linear

equations for the n + N- 1 parameters. The periodici ty condition s (i) (0) =

= s (i) (Nh) (i 0, 1, .•• , n-2) yields n- 1 addi tional linear equations. Con­

sequently, the sol vabili ty of an (n + N- 1) x (n + N - 1) linear sys tem of

equations must be examined. It is well known that a unique solution exists

if and only if the corresponding homogeneaus system only has the trivial

solution. Therefore one has to prove that the null function is the only

function in S (pn,h,N) satisfying s (Jlh + o:) 0 (\l E ~). In view of (3.3.1)

this amounts to showing that the difference equation

does not

the last

n-1 L av(a)s(\lh+vh+x) = 0

v=O

have a nontrivial Nh-periodic

few lines of the first proof)

0 s; x s h)

solution. However, this

from the fact that rn-1 \1=0

= Wz (pn;h,a) has, by assumption, no zeros at the N-th roots of

follows (cf.

av(a)z \1

unity. 0

86

Theorem 4.3.3 has an interesting consequence for differential operators

p (D) satisfying p*(D) = p (D) and such that pn only has real zeros. It n n n

then follows from (3.2.11) that

Hence, if nis even then ljl_1 (pn;h,h/2) =0 and in view of Lemma 3.2.9 the

function ljl_1(pn;h,•) has noother zerosin (O,h]. If nis odd, then

Hence ljl_1

(pn;h,h) = 0 and h is the only zero in (O,h] of w_ 1 (pn;h,•).

These observations combined with Theorem 4.3.3 lead to the following gener­

alization of Theorem 4.1.1.

THEOREM 4.3.4. Let Pn E 'ITn (n ~ 2) be a monic po'tyn.omial having on'ly real

aeros, and Zet p: = Pn· TMn for a E (O,h] and an N-periodic sequence

(y l"' there e:rists a unique periodic cardinal 1.-spUne s E s (p ,h ,N) ~ _." n

satisfying the inter-poZation property s (lJh +a) = y IJ (IJ E 'E) in each of the

following cases:

i) Nis odd, n E ~ arbitrary, a arbitrary.

ii) N is even, n is even, a f. !h.

iii) N is even, n is odd, a :{' h.

We emphasize the fact that case i) of this theorem holds without the condi­

* tion pn = Pn· This is an immediate consequence of Theorem 4.3.3; we state

it here as

COROLLARY 4.3.5. Let Pn E 1rn (n <!: 2) be a monic polynomiaZ having onZy reaZ

zeros and Zet N be an arbitrary odd integer. Then for any a E (0 ,h] and any

N-periodic sequence (yll)=~ there e~sts a unique sE S(pn,h,N} satistying

s ( ~h + a) = y lJ ( 1J .:: lt.).

87

Relation (3.3.3) holds for all s ~ S(pn,h) and a fortiori also for all

functions in Ker(pn). In view of Peano's remainder formula (cf. Lemma 1.4.5)

one has

(4.4.1)

0

nh

f K (nh x,y

for all f E AC(n) (~) and an appropriate kernel which, because of its depen­

denee on x and y, is denoted by K x,y In order to determine K we make use of (3.3.2) and take ~ x,y

0 in (4.4.1).

It then follows from Lemma 1.4.5 that

K (nh t) x,y

n-1 2 {ak{y)q>+(kh+x-t)- ~(x)q>+(kh+y-t)}

k=O

where, as usual, q>+ is the Green's function corresponding to pn(D).

Hence, for t E JR,

(4.4.2) K (t) x,y

n-1 2 {ak(y)qJ+(t- (n-k)h+x) -~(x)q>+(t (n-k)h+y)}.

k=O

Formula (4.4.2) shows that K is an t-spline defined on ~ having the x,y knots h-x,2h-x, •.• ,nh-x,h-y,2h-y, ••• ,nh-y. Using (3.3.3) and (4.4.2) one

can easily verify that the function K enjoys the following properties: x,y

r K (t) 0 x,y

(4 .4. 3) ") K (~hl = 0

::i)

x,y

K (t) = 0 x,y

(t t (O,nh]) I

(~ = Oll, ••• ,n) 1

(t E :R) , in case x = y .

Next we investigate the sign structure of K ; information about this and x,y properties (4.4.3) are needed in the sequel to derive an error estimate for

cardinal t-spline interpolation, which is the ultimate aim of this section.

Details about the zeros of K are given in the following x,y

LEMMA 4.4.1. Let the differential operator pn(D) (n ~ 2) be diaconjugate on

[O,nh]. If 0 < x < y s h then the function K as given by {4.4.2) has x,y

88

simple strong zeros in h,2h, ••• , (n-1)h. Moreover, these a'I'e the only zeroa

of K in (h-y,nh-x). x,y

PROOF. We first observe that i) of (4.4,3) implies that K (t) = 0 for x,y ti (h-y,nh-x). Furthermore, the number of knots of K in (h-y,nh-x) is x,y equal to 2n- 2. As [ 0 ,nh] is an interval of diseohjugaey for p (D) we may

[n-1] n write pn (D) = DnDn-l • • • D1 DnD , where the operators Di are given by

(1.4.2). Now

sinee an-l (x) = tp(x) :F 0 (ef. ii) of Lemma 3.2.7) and a0 (y) = w0 Cy) = p (O)tp(y-h) '# 0 (ef. (3.2.9) and (3.2.13)). By Theorem 1.4.23 (ef. pp.

n 14, 15) one has

(4.4.4) n-1 Z((D.l 1 ,K ,(h-y,nh-x)) s 2n-2- (n-1) = n-1. J. x,y

We reeall (cf. (4.4.3)) that K (jh) = 0 (j = l, ••. ,n-1). Soit remains to x,y prove that the points h,2h, ••• ,(n-1)h are strong zeros. In order to do so

we assume that ih, say, is not a strong zero. Since K is an l-spline we x,y then conclude that K vanishes in an interval between two conseeutive x,y knots. Aeeordingly, we may asssume that K (t) = 0 (ih-x S t S (i+l)h-y).

x,y ( ) Consequently, by (4.4.1), for all funetions f E AC n ([O,nh]) for whieh

pn(D)f vanishes identically outside the interval [(n-i-1)h+y,(n-i)h+x] one

would have

(4.4.5)

In view of (3.2.10) formula (4.4.5) may be written as

(4.4,6) n-1 ~ Bh (pn; y + (n-1-v)h) f(x + vhl =

vcO

n-1 ~ ~(pn; x+ (n-1-v)h)f(y+vh) •

v=O

Having es tablisbed this, our next purpose is to obtain a con tradietion by

exhibiting a funetion f E AC(n) ([O,nh]) for which pn(D)f vanishes outside

[ (n-i-1)h+y, (n-i)h+x], but for whieh (4.4.5) does not hold. To this end we

first assume that i~ n/2. As a consequence of Lemma 1.4.17 there exists

at least one function f 1 E Ker(p0

) such that f 1 (x +Vh) = 0

89

(V O,l, ••• ,(n-1-i)), f1

(y+Vh) = 0 (V 0,1, ••• ,n-2-i) and

f 1 (y+ (n-1-i)h) = 1. Here the assumption i 2:: n/2 is used to ensure that the

number of interpolation conditions for f1

is at most n. It is now easy to

construct a function f E AC(n- 1) ([O,nh]) that coincides with f1

on

[O,(n-i-l)h+y] and that vanishes on [(n-i)h+x,nh]. Substitution of this

function into (4.4.6) yields 0 = Bh(pn;x+ih}. However, this violates

(2.2.10) since 0 < x+ih < nh, and a contradietien is obtained. If i< n/2

then in a similar way we may prove the existence of a function f 2 with the

properties: pn (D) f 2 vanishes identically outs i de [ (n-i-1) h+y, (n-i) h+x],

is identically zero on [O,(n-i-l)h+y], f((n-i)h+x) ~ 1, f(x+vh) = 0

(V n-i+l, ••. ,n-1), f(y+vh) 0 (V= n-i, ••• ,n-1). Then (4.4.6) reduces

to Bh(pn;y+(i-l)h) = 0, which violates (2.2.10) and again a contradietien

is obtained. Consequently, the assumption on ih not being a strong zero

cannot be maintained. We conclude that jh (j = 1, .•. ,n-1) are strong zeros

and that these zeros are simple in view of (4.4.4). The last assertien of

the lemma fellows from the fact that K is an .C-spline, and therefore a x,y

zero of K is a strong zero or it belongs to an interval that is a strong x,y

zero.

If sE S(pn,h) interpolates a given function fat the points llh +a (Jl E ZZ),

then (4.4.1) combined with (3.3.3) implies that for ]l ZZ one has

(4.4. 7) r K {nh-T)p (D)f(]lh+T)dT. x,a n 0

Formula {4.4.7) may be used to obtain an estimate for lf(x) -s(x) I (x E :R).

For this purpose we assume in addition that llpn(D}fll < 00 and, furthermore,

that ZI-t ljiz (pn;h,a) has no zeros on the unit circle. Then (4.4. 7) can be

considered as a linear difference equation the salution of which may be

obtained in a similar way as in Lemma 3.4.1. Hence it fellows from (3.4.4)

that

nh

(4.4.8) f (ph + x) - s ( ]lh + x) J 0

"" K (nh-T) I c.p (D)f{j+p)h+T)dT

x,a j=-oo J n

where the coefficients cj are given by the Laurent expansion

(4.4.9)

c

90

that converges on a ring containing the unit circle in its interior.

Changing the order of summatien and integration and replacing T by T -jh,

we rewrite formula (4.4.8} for 0 ~ x~ h and IJ E <E in the form

"' (4.4.10} f(!Jh +x) - s (!Jh +x) I K (T)p (D)f(!Jh+T)dT x,a n

where

(4.4.11) K <tl := x,a

co

L c. K ((n+j)h- t) • j•-<» J x,cx

Formulas (4.4.10} and (4.4.11), together with i) of (4.4.3), immediately

yield

nh

(4.4.12) "' I f (!Jh +x) - s {!Jh +x} I ~ L I c .1 j=-"" J J IK (TlldT lip (D)fll • x,cx n

0

In order to use this estimate one has to evaluate the sum and the integral

in the right-hand side of (4.4.12}. With respect to the integral, we con­

sider the perfect Euler !-spline E(pn;h,•} (cf. Definition 3.2.12). In

order to ensure the existence of E(pn;h,•) it is necessary to show that

p (-1) 'I 0. However, if p (-1) = 0 .then a nontrivial function gE Ker(pn) n n exists such that g(t+h) = -g(t) (t E::R). Consequently, g would have more

than n- 1 zeros in (O,nh] which contradiets the fact that pn (D} is discon­

jugate on (O,nh]. Taking into account the sign structure of K N as given x.,-.. by Lemma 4.4.1, writing f 0 := ECpn;h,•), and using t 0 (t+h) =- f0 (t) (te :R),

we conclude from (4.4.1) that

0

nh nh

f IK (T) ldT x,cx I f K (nh-t)p (Dlf0

(T)dTI x,a n 0

lljlE(pn;h,a}f0 (1Jh+x) -ljlE(pn;h,xlf0 (1Jh+cxll =

lljl_ 1 <pn;h,cx)f0 (ph+x) -w_1 (pn;h,x)f0 (1Jh+all

It fellows from Definitions 3.2.5 and 3.2.6 that

nh

(4.4.13) J IK (1') jd1' "' lljl 1 (p ;h,cx) (fo(llh +x) - sf (ph+ x» I x,a - n 0

0

91

where

(4.4.14)

sf0

being the Euler .C-spline in S (pn 1 h) interpolating f0

at the nodes Jlh +a

().1 E JZ) •

The next step is to estimate E~ jc,j. Let z1

, ••• ,zn_1

be the zerosof the ]=-"" J

polynomial z~W2 (pn;h,a}. Then in view of property ii) of Lemma 3.2.7 one

has

n-1 ~(a) n

j=1

In case lz. I > we write ~

whereas lz. I < 1 gives rise to ~

1 1 co (z.)k --=-- 2 .2:. z1 - z z k=O z.

Furthermore,

lz. I- 1 l.

1

(z- z .) J

(z E Ç) •

( lz. I < 1) • l.

-1 Using these relations in the Laurent expansion (4.4.9) for 1jl2

(pn;h,a} we get

The results obtained above are combined to yield the following

THEOREM 4.4.2. Let f E AC(n) (lR} (n ~ 2} and let the ~perator pn(D) be

disoonjugate on [O,nh]. If a E (O,h] is suah that the potynomial

92

z..., ljlz (pn;h,a) has no zeros on the unit airale and if IJpn (D) fll < "'• then

there e~sts a unique sf € S(pn,h) suah that

{

i)

(4.4.16) ii)

(ll € ~) ,

(x <: JR) ,

where

I w_l(pn;h,a} I

K := n-1 I

ip(a) n {1- lz.l) j=l J

z1 ,z2 , ••• ,zn-l are the zeros ofz~ljiz(pn;h,al, f 0 is the perfeat Euler

l-spline aorresponding to pn(D), Sfo is given by (4.4.14) and where 1p is

the fundamental jUnation aorresponding to pn(D).

PROOF. Formula (4.4.8) can be used to define a function sf having the

desired properties i} and ii). The uniqueness of sf follows from iil 1 since

otherwise there would exist a bounded nontrivial function s € S(pn1hl with

s(llh +a) = 0 (ll <: ïZ) which1 by Lemma 3.4.1, violates the assumption on the

zerosof z~ 1/Jz(pn;h,a}. This proves the theorem.

If pn has only real zeros then, 'according to Lemma 3.2.11, z~-+ ljlz(pn;h,a)

has only nonpositive zeros and hence

I n-1 I I n-1 I lp(a) n o-lzjll = lp(a) ." (1 +zj)

j=l J=l

Taking into account (4.4.15) and using (4.4.9) one obtains

Consequently, cjcj+l :S 0 (j .:; ~) and

co

.l Jcjl = lw_ 1 <pn;h,all-1

• J=-00

Inequality ii) of (4.4.16) may then be written in the form

(4.4.17) I f(x) - s (x) I s I f 0 (x) - sf (x) 111 p (D) f 11 0 . n

(x <: JR) 1

0

93

where f 0 is the perfect Euler .C -spline E (p ;h, •) and sf is the Euler 1:.-n 0

spline given by (4.4.14).

Moreover, in this situation (4.4.17) is best possible, since the upper bound

is attained for the perfect Euler .C-spline f(p ;h,•). We state the result - n

obtained as

COROLLARY 4.4.3. Let pn E nn (n ~ 2) be a monic polynomiaZ having onZy real

zeros. If a E (O,h] is such that w_1

(pn;h,a) .Jo, then for every fE AC(n) (lR)

there exists a unique sf E s (pn,h) satisfying

(4.4.18) ~ ii) {.) f(llh +a) (ll € 1::) ,

(x E JR) ,

where t 0 is the perfect Euler 1:-spZine f{pn;h,·) and sf0

is the Euler .C­

spZine interpoZating f 0 at the nodes llh +a (IJ E E).

Moreover, the upper bound in (4.4.18) is best possibZe.

REMARK. For general operators pn(D) it is an open problem todetermine a

best possible upper bound for I f(x) - sf(x) I (x E JR); to achieve this

detailed information about the zerosof z~ lj!z(pn;h,a) would be needed.

This sectien is devoted to the problem of periodic cardinal .C-spline inter­

polation, with emphasis on error estimates. If f E AC(n) (JR) has period Nh

and if z~lj!z(pn;h,a) has no zeros at the N-th roots of unity, then by

Theerem 4.3.3 there exists a unique sf E S(pn,h,N) interpolating fat the

nodes 1-lh +a (J.l E :El. Now the problem arises to compute for all x E JR

(4.5.1) sup {I f(x) - sf(x) I I f E AC(n) (lR), f is Nh-periodic} llpn (D) fll :>1

If lj!z(pn;h,a) has no zeros on the unit circle, then (cf. Theerem 4.4.2) the

unique bounded cardinal .C-spline sE S(pn,h) interpolating fat the nodes

1-1h +a has, due to i ts unie i ty, period Nh and an estimate for ( 4. 5 .1) is

given by ii) of (4.4.16). In case pn has only real zeros Corollary 4.4.3

94

applies and the estimate (4.4.18) may be used. However, ii) of (4.4.18) is

best possible for cardinal !-spline interpolation and it will be best

possible for periodic cardinal !-spline interpolation only if E(p ;h,•) has n

period Nh. Since E{p ;h,t+h) =- E(p ;h,t) (t E :R) one has E(p ;h,t+Nh) n n . n

= {-1)N E(p ;h,t). Consequently1 E(p ;h 1t) has period Nh only if Nis an n n even number. This gives rise to the following

THEOREM 4.5.1. Let p e ~ (n ~ 2) be a monia polynomiat having only real n n

zeros and Zet N € ~ be even. If Ijl 1 (p ;hl a) -1 o, then for every. Nh-periodia - n

funation f € AC(n) (JR) there exists a unique sf € S(pn1h1Nl suah that

{

i)

(4.5.2) ii)

(IJ € ~) I

(x € :R) 1

where t 0 is the perfeat Euler !-spline E(pn;h~·> and Sfo is the Euter t­

spZine in S(pn,h1Nl interpoZating f0

at the nodee Jlh+a (\l E 1/.), Moreover,

iil of (4.5.2) is best possible.

* If pn has only real zeros and if1 moreover, pn = Pn then ljl_ 1 (pn;hla) = 0

fora= h/2 and neven, and w_ 1 (pn;h 1a) = 0 fora= hand n odd (cf. p. 86).

Consequently, for operators p (D) for which p (D) = p*(o), Theerem 4.5.1 n n n furnishes the following corollary.

COROLLARY 4.5.2. Let pn E nn (n ~ 2) be a monia poZynomiaZ having only real

zeros with p: = Pn and Zet N E ~ be even. Then for every Nh-pePiodia funa­

tion f E Ac(n) (lR) there exists a unique sf E S(pn1h,N) interpoZating fat

the nodes \lh +a (IJ e tl.) in eaah of the foUowing aasea:

i) n is even, a -1- !h, ii) n is odd, a -1 h.

Moreover

{4.5.3) (x € :R) ,

where f 0 is the perfect EuZer t.-apZine f(p ;h,•) and Sf ia the EuZer !-n 0

spZine in s (pnlh,N) interpoZating ! 0 at the nodes ph+ a (p € tz.).

InequaZity (4.5.3) is best poasibZe.

95

We now return tothe.general problem of estimating (4.5.1) 1 only assuming

that z + ~z(pn;h1al has no zeros at the N-th roots of unity. As in the case

of cardinal !-spline interpolation (cf. (4.4.10)) we shall determine a

kernel function K such that for all x € [0 1h] and V € ~~ x 1a

(4.5.4) fNh K (t)p (D)f(Vh +t)dt x1a n

0

Let <\ the k-th component of the vector d E :nl be given by

nh

(4.5.5) ~ := f K (nh-t)p (D)f(kh+t)dt x 1 a n (k 0 I 1 I ••• I N-1) ,

0

where K is given by (4.4.2) I let ef = f-x 1 a T

•• ,ef(x+(N-1)h)) • Using (4.4.7) for ll = 0,1, ••. ,N-1 one obtains, similar to

(4.3.4), the system of linear equations

(4.5 ,6)

Since ~ (p ;h,a) ~ 0 if zN = 1, the circulant matrix C has an inverse z n denoted by C(b0 ,b11 ... 1bN_

1) (cf. p. 84}. Hence for ll 011, ... ,N-1 and

0 :> x:> h one has (cf. (4.3. 7))

(4.5.7) K (nh-T)p (D)f(jh+t)dT 1 x,a n

where (bll1 0

, ... 1 bll1 N-l) is the (ll+1) -st row of C (b0 1b11 ... 1bN_1) •

Let the function K be defined by x,a

(4.5.8) K (t) := x,a I k=-"'

K (nh - t - kNh) x,a (t E JR) •

Evidently1 K is an Nh-periodic function. Since f is also an Nh-periodic x1a

function1 the integral in the right-hand side of (4.5.7) can be written as

b . K (nh-t)p (D)f(jh+t)dt \l 1 J x,a n

96

"" (k+l)Nh N-1

~ J ~ bl-1 0

k=-ro j=O ,J K (nh-t)p (D)f(jh+t)dt

x,a n

~ k=-""

kNh

Nh N-1

f I b 0

j=O 1-l,J 0

Nh N-1 ~

K (nh-T-kNh)p (D)f(jh+T)dT = x,a n

= J ~ b 0

j=O 1-l,J 0

K (t)p (D) f(jh + T) dT x,a n

Replacing t by t- (j-]l)h and taking into account i) of (4.4.3) we obtain

00

J N-1 I b K (t- (j-]l)h)p (D)f(]lh+T)dt

j=O ]l,j x,a n

The required kernel function K will now he defined by (cf. (4.5.4)) x,a

(4.5,9) K Ctl x,a

N-1 := I b 0 i< ct- (j-lllh>

j=O 1-l,J x,a (t E JR) ,

We cbserve that K does notdepend on 1-1, since (b ~, ••• ,b N 1l is the

x,a 1-l,v ~ ll, _ {\.1+1)-st row of the circulant matrix C(b0 , ••• ,bN_

1> and Kx,a is periodic

with period Nh. Hence

N-1 {4.5.10) K <tl x, a ~ bj i< <t- jhl

j=O x,a (t E JR) 1

and (4.5.7) may he written as

Nh

ef(l-lh+x) = I K (t)p (D}f(llh+t}dt, x,a n

0

where 0 ~ x ~ h and ll E lZ •

The results obtained are collected in the following lemma.

LEMMA 4.5.5. Let a E (O,h]~ Zet p E ~ (n ~ 2) be a monie poZynomiaZ and n n Zet N E :tt. If Wz (pn;h,a) .; 0 for zN = 1~ then the unique funation

s f E s (p n, h, N) intertpo Zating an Nh-periodia funation f E AC ( n) ( JR) at the

nodes )Jh +a (ll E '3.) satisfies for 0 ~ x ~ h the reZation

(4.5.11) f(llh+x) -sf(llh+x)

where K is given by (4.5.10) x1a

0

Nhf K (T)p (D)f(llh+T)dT X 1 il n

97

In order to obtain (4.5.1) one has to determine the supremum of the integral

in the right-hand side of (4.5.11) over all Nh-periodic functions

f E AC (n) (JR) for which 11 p0

(D) f 11 "; 1. With respect to this problem we need

the following lemma.

LEMMA 4.5.4. Let FE C([0 1 T]), where T is a positive number. If there exists

a T-periodia funation g0 E Ker(pn) coinaiding with F in at most finitely

many points and suoh that for all T-periodic functions g E Ker(pnl

(4.5.12)

then

r g(T) sgn(F(T) -g0 (T))dT

0

Û 1

(4. 5 .13) sup { I F ( T) pn (D) f (<) dT I f E AC (n) (JR) , f is T-periodic} 11 p

0 (D) fll ";1 O

T

J F(tlpn(Dlf0 (<)d< I

0

Mhere f 0 E AC(n) (JR) is a T-periodic function satisfying the differential

equation

sgn (F (t} - g0

(tl l (0 < t < T) •

PROOF. Im Lemma 6.2.1 of Chapter 6 it will be shown that the differential

equation p (D}f(t} n ( ')

satisfying f J (0}

property that

T

u(t) (0 < t < T), where u EL ([O,T]}, has a salution

f(j} (T) (j = 0,1 1 ... ,n-1) if :nd only if u has the

(4.5.15) I g(T)U(T)dT 0

0

for all T-periodic functions g E Ker(p0).

98

Let u be the set of functions u e L.,. ([O,T]) satisfyin9 (4.5. 5) and

11 uii[O,T] :;; 1. It fellows that the supremum in (4.5.13) equals

T

(4.5.16) sup {I F(t)u(t)dT I u e u} 0

Obviously, for all T-periodic functions g e Ker (pn) one bas

T T T

IJ FCrluCrldri==II<FCr)-g(r))uCtldrj:;; J!F<r>- 9 cr>ldr

0 0 0

Hence the supremum in (4.5.16) is bounded from above by the Lj1-distance of

the continuous function F to the finite dimensional space of functions

9 e Ker(p0

) that are T-periodic.

According to a general cbaracterization theorem for L1-approximation (cf.

Cheney [12, p. 220]), a function 90

tbat coincides with Fin at most

finitely many points and that satisfies (4.5.12), is a best L1-approxima­

tion toF, i.e.,

T T

J IF(T) -g0 Cr> ldr:;;

0

J IF(T) .. g(T) !dt ,

0

for all T-periodic functions 9 E Ker(pn).

Therefore, by taking u0

:= sgn(F- g0), it fellows that u0 E U and thus for

all u e u

T

I I F(T)U(T)dT' :;;

0

T

J !F(T) - g0 (T) jdT

0

= f(F(T) -g0 (t})u0 (T)dT = J F(T)UO(T)dT

0 0

By taking f0

€ AC(n) (JR) to be the T-periodic function tbat sátisfies

pn(D)fo(t) = uo(t) (0 < t < T), tbe lemma is obtained. D

REMARK. If F bas not more than finitely many points in common.with any T­

periodic function gE Ker(pn)' then, according to Schönhage [55,p. 169], a

function g0 satisfying (4.5.12) always exists.

99

Lemmas 4.5.3 and 4.5.4 easily yield the following theorem.

THEOREM 4.5.5. Let a E (O,h], Zet p E TI (n ~ 2) be a monic poZynomial and n n N

let N E N be such that ~ (p ;h,a) ~ 0 ror z z n 1. Furthermore, let x E [ 0 , Nh J

and X 1 := x- Vh E [O,h] UJ{th V E :N0

.

If there exists an Nh-periodic funetion g0 E Ker(pn)

with K (ef. (4.5.10)) at at most finitely many x,a moreover, suoh that

T

(4.5.17) f g(1:) sgn(K , (1:)- g0

(1:) )d1: x ,a 0

0

for all Nh-periodic functions gE Ker(pn), then

Nh

suoh that g0 ooincides

in [O,Nh] and,

(4.5.18) I f(x) (x) I s I IK, <1:> -g0 CTJ ld-r lip (Dlfll , x ,a n 0

where sf E S(pn,h,N) is the unique function interpoZating the Nh-periodio

function f E AC (n) (JR) at the nodes )lh +a (ll E 'tl.). Moreover, the upper

bound in (4.5.18) is best possible.

It seems difficult to obtain detailed information about g0 when dealing

with a general differential operator pn(D). As a consequence, it is not at

all easy to evaluate the right-hand side of (4.5.18). Even for periodic

piecewise linear interpolation, i.e., p2

(D) = o2 , the problem is far from

simple. This will be apparent from the following

EXAMPLE. We consider the problem of periadie piecewise linear interpolation,

i.e., p 2 (D) n2 • Then (cf. (3.2.21) and (3.2.18))

tz +h- t

Takinga = h/2 and N odd we may apply Theerem 4.3.4, which ensures the

existence of a unique periadie linear spline sf E S(p2 ,h,N) interpolating a

given Nh-periodic funetion f E AC (2) (JR) at the midpoints of the mesh in­

tervals. Note that $_ 1 (p2 ;h,h/2) = 0.

The function Kx,h/2 (cf. (4.4.2)} is now given by

' (4. 5.19) Kx,h/2 (t)

h 3 h h 2(t-2h+x)+-(h-x)(t-2h)++2(t h+x)+-x(t 2)+

100

Substituting x = h in (4.5.19) we get a so-called roof function

(4.5.20}

the graphof which is given in Figure 1.

h'/4--"-----D 0 h~/~2----~h---------_.--t

Figure 1.

'If we takeN= 3, the circulant matrix A (cf. (4,3.4}) equals ! hC(1,1,0},

and its inverse is h- 1 C(1,-1,1).

In view of Formulas (4.5,8} and (4.5.9) the kemel function ~,h/2 is in

this particular case given by

(4.5.21) - -1 ~ ~ ~

~.h/2 (t) = h {~,h/2 (t) - ~,h/2 (t- h) + ~.h/2 (t- 2h)} ,

where

~

~,h/2(t) L: ~.h12 ( c2- 3k>h- t> k=-00

(t € :R) ,

Taking into :ccount the graphof ~.h/2 as given in Figure 1, we easily

verify that ~,h/2 is a periodic function with period 3h, the igraph of

which on the interval [0,3h] is shown in Figure 2.

0 h ~h 2h -t

Figure 2.

In view of (4.5.21) and Figure 2 the graph of the 3h-periodic function

~,h/2 may now be easily obtained; cf. Figure 3 fora sketch on [0,3h].

101

0 __,._ t

--- h/4

Fig-ure 3.

Thenext step is todetermine the function g0 as given by {4.5.17). Taking

into account that g0 ~ Ker(D 2) and that g0

is 3h-periodic, we conclude that

g0 is a constant function, g0 {t) "'c say, where by (4.5.12) c is such that

r sgn(~,h/2 (T) -c}dT

0

0

One easily finds that c = h/16. In view of the graph of ~.h/2 as given in

Figure 3, one then obviously has on (0,3h)

-1 ( 0 < t < ir h) 1

sgn(~,h/2(t) - :6) -1

7 < t < 8 h> I

h < t < 2.h) 8

(2.h < t < ~h) 8 8 I

-1 h < t < 3h} .

Hence (cf. (4.5.18))

3h

I f(h) - (h) I :;;; f ~~~h/2 (T)

0

1~ I dT 11 f"ll = ;; h 2

11 f" 11

The function f 0 (cf. Lemma 4.5.4) for which the inequality above reduces to

an equality is not an Euler !-spline corresponding to the operator 3

Dp 2 (D) = D 1 but a perfect ..C-spline (cf. Definition 1.3.5), since

Dp 2 (D) f 0 jumps from ± 1 to + 1 at the nonequidistant points h/B1 7h/8, 9h/81

15h/8.

102

5. ON THE LANVAU PROSLEM FOR SECONV ANV THIRV ORVER VIFFERENTIAL OPERATORS

5.1. Inttoduction and éumm44y

As pointed out in the general introduetion this thesis consists of two

parts. Here we begin the secend part, which deals with the Landau problem

for linear differential operators. The name "Landau problem" originates

from a few interesting inequalities, published by Landau as early as 1913,

connecting llfll, llf' 11, and llf" 11. His main results are contained in the fol­

lowing theorem (cf. pp. 3, 4 for notation).

THEOREM 5.1.1 (Landau [30]). Let f E w< 2> (lR) be such that llfll < oo. Then

llf' 11 ::;; 12 fllfllllf" 11 •

FUPther, Z.et f E w< 2> (lR~) be such that llfll+ < "" Then

The constante 1:2 and 2 in these inequaZ.ities are best possibZ.e, i.e., they

aannot be repZ.aaed by smalter ones.

Later, in 1939, Kolmogorov generalized the first part of Landau's theorem

to higher derivatives. His result reads as fellows.

THEOREM 5.1.2 (Kolmogorov [27]). Let f E w(n) (lR) (n ~ 2) be suah that

llf 11 < ""· Then

(5.1.1)

~here the aonstants c k given by n,

C := K /K(1-k/n) n,k n-k n

~ith

(k = 1, 2 1 • • • 1 n-1 ) 1

4 L 1 i+1 (2j + 1) i+1 1T j=O

(i even) ,

K. :!;';';' ~

00

(-1) j 4 L i+1 (2j+1)i+1 1T j=O (i oddJ ,

are best possibZe.

Moreover, Kolmogorov established that the extremaZ funotions of (5.1.1},

i.e., those functions for which (5.1.1) reduces to an equality, are the

Euler splines. Besides these two theorems it is appropriate to mention an

extension of Landau's results to higher derivatives for functions (n) +

f E W (1R0) , due to Schoenberg and Cavaretta [ 54]. The emphasis of

their workis on the construction of the extremal functions of (5.1.1). + Upper bounds for cn,k in (5.1.1) with respect to functions defined on ~O

are given by Ste~kin [60].

103

2 3 Fora lucid exposition of Landau's problem for the operators D and D the

reader is referred to Schoenberg [52].

In order to put these problems into a more general context we define the

set of functions Fm(pn,J) by

(5.1.2)

here pn E 1Tn is a monic polynomial, J is a closed subinterval of ~. m is a

positive number, and the set W(nl (J) is defined on p. 3.

In general, a Landau problem can then be described as follows: determine

the best possible upper bound for llpk(D)fiiJ on Fm(pn,J), where pk(D) is a

linear differential operator of order k n-1. Functions for which this

upper bound is attained are again called extremal functions. Recently,

Cavaretta [ 11] established · that Euler splines are extrema! for the

problem of maximizing ll(aDi+bDi+llfll on F {Dn,m.), where a and bare arbi-m

trary real numbers and 0 s i s n-2. For a result invalving a finite inter-

val J the reader is referred to Chui and Smith [13], who compute the maxi-2

mum of llf' liJ on Fm (D ,J) for an interval J of arbitrary length. Using the

approach of Cavaretta [9], occurring in an elementary proof of

Kolmogorov's Theorem 5.1.2, Sharma and Tzimbalario [58] investigated the

following special Landau problem: if pn E 1Tn is a monic polynomial having

only real zeros and if pk is a divisor of pn, determine the maximum of

104

llpk(D) fll on F (p ,m). It turns out that a perfect Euler .C-spline is ex-m n tremal for this problem. The basic idea of Cavaretta in his elementary

proof of Theerem 5.1.2 is first to prove that the Euler splines are extremal

with respect to the periodic functions in F (Dn,JR}, and then to approximate m a function f € F (Dn,lR) by periodic functions.

m

The contents of this chapter may be roughly described as fellows. In Sectien

5.2 the significanee of the Landau problem with respect to optimal differen­

tiation algorithms is briefly explained. Our approach to studying the Landau

problem is based on an investiqation of the set rm(pn,J,E;) of lRn defined by

(5.1.3)

where Fm(pn,J) is given by (5.1.2) and ~ e J.

By taking E; = 0 and J lR or J = lR~ the problem of maximizing 11 pk (D) fll on

Fm{pn,J) is equivalent to the problem of maximizing a linear function on

rm (pn,J ,0).

In Sectien 5.3 general properties of the sets Fm(pn,J) and rm(pn,J,~) are

derived. For instance, with respect to Fm(pn,J) it is shown that any

sequence of functions in Fm(pn 1 J} contains a subsequence that converges to

an element of Fm(pn 1 J), uniformly on each compact subinterval of J. More­

over, simultaneous uniform converganee occurs for the successive derivatives

up to the (n-1)-st order. The set rm(pn,J,E;) is shown to be a convex and

compact subset of lRn having ~ as an interior point. Particular attention

iS gi Ven tO the SetS r (p 1 m, 0) and r (p 1 m~ 1 0) , The importance Of the Sets + n n

r(pn,lR,O) and r(pn,m0,0) for the Landau problem is explained in Sectien

5.4. In Sectien 5.5 the Landau problem for secend order differential opera­

tors is studied in detail. With respect to the full-line case (J =lR) our

main result is a characterization of the boundary of the set rm(p2,m,O) in

terms of perfect Euler .C-splines (Theorems 5.4.3 and 5.4.4). A few examples

are given to illustrate the theorems. In the half-line case (J =lR~) it is

shown that the role of the perfect Euler .C-splines is taken over by the so­

called one-sided perfect Euler .C-splines, which are used to parametrize the + boundary of the set r m Cp2, m0 , 0) • Various cases wi th re gard to the location

of the zerosof p2 are considered. Problems,analogous to those of Sectien

5. 5 for general third order differential operators are much more difficult

to deal with. Therefore, Section 5.6 is restricted to third order differen-

* tial operators p3 (D) with the property p3 (D) = p3 (D). Operators of this

105

3 kind have the form p 3 (D) = D +cD. Only the full-line case is studied for

these operators under the additional assumption that c > 0 (this last

restrietion is not essential). The half-line case for operators p3

(D) of

the indicated form is, up to now, unsolved. In Chapter 7 the half-line case 3

for the operator p3

(D) = D is discussed in detail.

The purpose of this sectien is to outline briefly how Landau's problem is

related to the construction of optimal differentlation algorithms. This

problem occurs in the theory of optimal recovery (cf. Michelli and Rivlin

[39, p. 27]} in conneetion with the approximation of a differentlal operator

by means of bounded operators (see also Ste~kin [59]).

Let f E W (n) (]R) (the set of functions f E AC (n) (lR) for which llf (n)ll < oo) •

The objective is to construct an optimal algorithm for the computation of

pk (D) f (0) (1 :> k :> n-1) , based on the fact that llpn (D) f 11 :> 1 and that

required function values of f can be computed with an error bounded by €.

To be more precise: let A be any mapping from L00

(lR), thesetof essentially

bounded measurable functions into lR, and let g E L~(lR) be such that

lig fll < € for a given € > 0. Then A is interpreted as an algorithm and

A(g) is considered to be an estimate for pk(D)f(O). The error in using

algorithm A is defined by

The intrinsic error is defined by

(5.2.1) Ê(E:) := inf EA (t:) , A

and Ä is called an optima'l a'lgorithm if EÁ (€) Ê(t:). The following lemma

shows how the intrinsic error Ê(t:} is related to Landau's problem.

LEMMA 5.2.1. The intrinsia error Ê(t:) as defined in (5.2.1) satisfies the

inequa'lity

where (pn,lR) is given by (5.1.2).

106

PROOF. For a function f E Fe: (pn ,lR) and any A ene has

and, since - f E Fe:(pn,JR), one also has

He nee

and therefore

(5,2.2) sup {pk(D)f{O) I f E F (p ,JR)}::; Ê(e:). e: n

In the following sectien it will be shown (cf. Lemma 5.3.1) that the func­

tional fl.-,..pk(D)f(O) is bounded on Fe:(pn,JR). Since for every f E Fe:(pn 1 JR)

and every a E lR the function t .....-r f ( t - a)

clude that the left-hand side of (5.2.2)

also belengs to Fe: {pn,JR) we con­

equals sup {flpk{D) f[l I f E F (p ,lR)}. e: n This proves the lemma.

In the specific case that pk(D) = D and that pn has only real zeros,

Michelli [38] bas obtained optimal differentiation algorithms. It turns out

that these differentiation algorithms are exact for perfect .f-splines eer­

responding to the operator pn(D) and, consequently, they are of the type

considered in Sectien 3.4.

In this sectien the sets Fm(pn 1 J) and rm(pn 1 J,~) as defined by (5.1.2) and

(5.1.3) will be considered. With respect to Fm(pn,J) a few results will be

given, the first of which reads as fellows.

LEMMA 5.3.1. For everry polynorrriaZ pk E ;rk (0::; k::; n) the functional

f-llpk(D)fiiJ is bounded on Fm(pn,J).

' PROOF. Let 0 < x1 < x2 < ••• < xn be such that pn(D) is disconjugate on

[O,xn]. We reeall (cf. the proef of Lemma 1.4.18) that the functions

t...._.cp(t-x1 ) {i= l, ••• ,n), where cp is the fundamental function correspond-

0

107

ing to pn(D), forrn a basis for Ker(pn). Since pk(D)~ E Ker(pn)' there exist

uniquely deterrnined coefficients a 1 , ••• ,an such that

pk(D)fl)(t) n L ai fl)(t-xi)

i=l (t EJR).

As a consequence of Peano's remainder formula (cf. Lemma 1.4.5) it fellows

that

pk (D) f(t) n !aif(t-x.)+

i=1 l.

t

f K(t-T)pn(D)f(T)dT

t-x n

where the kernel function K is given by

Hence

n L la. l!lfll +

i=1 l. J

(t E JR) •

xn

J IK(T) I d<llpn (D) fiiJ

0

In view of (5.1.2) thisproves the lemma.

We note that a proef of this lemma can also be found in Michelli [38]; how­

ever, there it is assumed that pn has only real zeros.

Our next result concerns a compactness property of the set Fm(pn,J).

LEMMA 5.3.2. Any sequence of funations (fk>7 in Fm(pn,J) aontains a sub­

sequence {fk )00

that converges toa funotion f E Fm(pn,J} suoh that j 1

(i 0,1, .•• ,n-1),

uniformly on eaoh oompact subset of J.

PROOF. It fellows from Lemma 5.3.1 that ll<i)!IJ ~ c (i 0,1, ••• ,n;

0

k = 1,2, ••• ) forsome constant C ~ 0, independent of k and i. By Ascoli's

theerem (cf. Kreyszig [28, p. 454]) there exists a subsequence (fk .l 7 and a func-• (n) (i) {i) J

t1.on f E AC (J) such that fk. + f {j + w) for each i= O,l, •.. ,n-1, J

uniforrnly on each compact subset of J. Since llfkjiiJ ~ m one has llfiiJ ~ m.

108

It remains to prove (cf, {5.1.2)) that llpn(D)fiiJ s 1. For almost every t.;: J

we have

n-1 f(n) (t) + L a f(i) (t)

i=O i

{ n-1 }

lim - 1- f(n-l) (t)- f(n- 1) (/';) + (t- 1;) L a· f(i) (t) !;+t t- ç; i"'O i

Hence

1 . 1 1' J.m t _ ç; J.m (4t j-+-oo

n-1 ) + L a1 f~~) (t) dT =

i=O J

t

lim t ~ " lim J {P (D) fk (T) "+t " '-+<x> n . "' J ç; J

n-1 } + L a

1 (f~i) (t) -<i) (T)) dt

i=O j j

Since for all t E J and almest every T " J

S l+M(t-T)

forsome constant M > 0, we conclude that lP (D)f(t) I s 1 (a.e.), hence n

Having established these properties for the set Fm(pn,J), we now turn to

rm{pn,J,() as defined in (5.1.3).

THEOREM 5. 3. 3. The set r m (p n, J ,I';) is a convex and compact subset of lRn

having Q as an interiol' point.

PROOF. Since Fm(pn,J) is convex the set rm(pn 1J,I;) (cf. {5.1.3)) is a1so

convex. Furthermore, the compactness property immediately fellows from

Lemma 5.3.2. In order to prove that rm{pn 1 J,I';) has Q as an interlor point

it suffices to consider the full-line case since rm(pn,JR,() c rm(pn,J,()

for all subintervals J cJR. Moreover, it is easily seen that rm(pn,JR,() =

= rm(pn,JR,O) for all l; € lR. Consequently, we have to prove that Q is an

interior point of rm(pn,JR,O). To this end, let the functions fi

(i= O,t, ••• ,n-1) be defined by

(t " lR) ,

D

where a is a positive number, sufficiently smal! to ensure that

fi Fm(pn,JR) (i= O,l, ••. ,n-1). Then

(fi (0) tfj_ (0) 1 • • • 1

where is the i-th unit vector.

109

Hence, ± a~i E fm(pn,JR,O) for i O,l, •.• ,n-1. Using the convexity of

fm(pn,JR,O) one easily concludes that 0 is an interior point. This proves

the theorem.

Obviously, fm (p ,J,~) c fm (p ,J,~) if m < m2 . In the following lemma a 1 n 2 n 1

condition is given ensuring that rm1 (pn,J,~) is a proper subset of

rm2(pn,J,~).

LEMMA 5.3.4. Let m1 < m2• there exists a nontri~ial funation f E Ker(pn)

that is bounded on J, then rm1

(pn,J,sl is a proper suhset of rm2 <pn,J,~).

(n-1) T n PROOF. We consider a point~= (f(~),f'(;), .•. ,f (;)) E lR, where

f E Ker(pn) is a nontrivial function bounded on J. Since Q is an interior

point Of fm1

(p ,J,;), there exists a positive À SUCh that E ctfm (p ,J,;). n 1 n

Furthermore, there is a positive ll such that llllfiiJ m2 - m1 > 0. It is easy

toverifythat (À+ rm (p ,J,I;) and (À+\lletifm (p ,J,I;), i.e., 2 n - 1 n

rml (pn,J,I;) is a proper subset of rm2(pn,J,I;).

REMARK. If all functions in Ker(pn) are bounded on J, then we can prove the 0

strenger assertien that rm1 (pn,J,I;) c rm2 (pn,J,~). This, of course, wil!

always be the case if J is a bounded interval. Furthermore, rm {p ,JR,~) is 1 n

a proper subset of fm (p ,IR,!;) if pn has at least one zero on the imaginary 2 n

axis, since then there is a nontrivial bounded function in Ker(pn). If there

exists at least one zero of pn with a nonpositive real part, then there is a

nontrivial function in Ker(pn) bounded on ~~ and thus Lemma 5.3.4 may be + + applied to guarantee that rm (p ,JR0 ,1;) is a proper subset of fm (p ,JR0 , ~).

1 n 2 n

From now on we restriet ourselves to the full-line case and the half-line + case, i.e., to J JR and J ~0 • Since fm(pn,JR,O) = fm(pn,JR,~) (I; E JR) and

D

110

+ + fm(pn,JR0 ,0l = rm(pn,JR0 ,~> (~ ~ 0), the notations will be shortenedas fel-

lows:

(5.3.1)

It is interesting to note that if pn has no zeros on the imaginary axis,

then a number Mpn exists such that lip (D)fll S: 1 and llfll <"" imply llfll s: Mp • n . n This will be provedinLemma 5.3.5. In order to arrive at this 1 result we

give the following definitions.

For n ~ 2 and p E n a monic polynomial not having purely imaginary zeros, n n -1

wedefine the function fPn astheinverse Fourier transferm of pn (iw), i.e.,

"' (5.3.2) 1

Fpn (t) := 21T _e __ dw

f iwt

Pn (iw) (t E R) •

Using the well-known formula

"" e dw

J iwt

(iw +a)k+l (t E R ; Re a > 0 ; k E :NO) t

one easily verifies that the function Fp decreases exponentially as n

ltl +co, and so the integral in the right-hand side of (5.3.3) is finite. In

conneetion with Fpn the number Mpn is now defined by

(pn has no zeros on the imaginary axis),

(5.3.3) Mpn

(otherwise).

LEMMA 5.3.5. If f E Fm(pn,JR) forsome m~ then llfll s: Mpn• where Mpn is

given by (5.3.3).

PROOF. There is nothing to prove if pn has a zero on the imaginary axis,

since then Mpn = ""· So we may assume that Pn (iw) :1 0 (w E R). We note that,

apart from the trivial function, every function in Ker(pn) is unbounded on

JR. Consequently, a bounded solution of the differentlal equation pn (D) f =u

is uniquely determined and it can be represented as a convolution integral

f(t) I Fpn(t-T)u(T)dT.

Therefore, we may conclude that for f E Fm(pn,lR)

"' (5. 3 .4) f (t) I Fp (t-T)p (D)f(T)dT n n (t E P) •

Hence, by (5.1.2),

PROOF. Lemma 5.3.5 implies (cf. (5.1.2)) that Fm(pn,lR) FMPn(pn,lR)

thus (cf. (5.1.3) and (5.3.1)) fm(pn) c~rMPn(pn). Moreover,

f (p ) c r (p ) if m ~ Mpn. Mpn n m n

111

and

In the half-line case there are no nontrivial bounded functions in Ker(pn)

if all zerosof pn have positive real parts. If this occurs, the function

Fp as defined in (5.3.2) vanishes for t > 0. Consequently, a function n

+ f E Fm(pn,lRO) can be represented as

"' (5. 3. 5) f(t) I Fp (t-T)p (D)f(T)dT n n

(t ~ 0) •

t

This leads to the following

LEMMA 5.3.7. IfaZl zerosof pn have positive real parts, then

PROOF. It fellows from (5.3.5) that

jf(t) I :'> J IFpn(t- T) laT

t

0

112

Consequently, r~(pn) = r~n (pn) (m ~ Mpn). Obviously, rMPn (pn) c r:Pn (pn).

If.!! E r~ (pn) then a function f E FMPn (pn,JR~) with.!! .. (f(O), f' (0), ••• I

f(n-l) (O))T can easily be extended to a function f <:: FM (p ,JR) by defining Pn n

pn(D)f(t) := 0 for all t < 01 sof is then given by (5.3.5) for all tE~. + . +

Hence, a. E r Mpn (pn) I therefore r Mpn (pn) :;) r Mpn (pn) I and thus r Mpn = r Mpn (pn). 0

5. 4 • On the Jtel.a.;ti.o n between the La.ndau. p!tob!em a.nd the .& e.U ar (p l a.nd m n ar+<P l

m n

T n We reeall {cf. Lemma 5.3.1) that for any ~ = cs0 ~a 1 , ••• 1Bn_ 1> E ~ the

functional f~-+IIS0f+B 1 f• + ••• +B 1f(n-l)ll is bounded on F (p ,JR).

n- m n Furthermore, one easily verifies that

(5.4.1) ' (n-1) 11 sup IIB0f+B 1f + ••• +Bn_1f fEF m (pn' JR)

sup CS0f(OJ +13 1f• (0) + ••• +Bn_1f(n- 1) (0)} fEFm(pn 1 JR)

It fellows from the definition of the set rm(pn) (cf. (5.3.1) and (5.1.3))

that, in order to obtain the supramum in (5.4.1), one has to compute the

supremum of the linear function x ...... aT x on r (p ) • Since r (p ) is convex - - m n m n

and compact this supramum will be attained at a point of arm{pn).

On the other hand, if ,!! E of m (pn) then in view of Theerem 2 in Luenberger

[31, p. 133] there is a hyperplane H containing.!! such that arm (pn) lies on one

si de of H. Hence a vector B <:: ~n exists such that 8 T x $ 8 Ta. for all

.!5. E rm(pn). Consequently, ;f fo E Fm(pn,JR) satisf;e;~6i) (ç;~ = a.i

{i Oll, ••• ,n-1) forsome ç; E ~, then f 0 is an extramal function (cf.

p. 103 for its definition), since in view of (5.4.1) the function f 0 maximizes 11 B

0f + 81f• + ••• + B

1f (n-l) 11 on F (p ,JR).

n- m n Summing up we conclude that a complete description of arm(pn) is intimately

connected with solving the Landau problem on ~. A similar conclusion holds

for the Landau problem on ~;. For n = 2 we are able to give such complete

* déscriptions, and for n = 3 under the additional condition that p3 = p3•

For the case n ~ 4, treated in Chapter 7, we investigate ()f~(pn) under the

conditions that pn has only real zeros and pn(O) = 0.

113

5.5.1. Introductory remarks

Taking into account the contents of Sectien 5.4 we aim at characterizing + the sets arm(p2) and arm(p2}. With respect to arm(p2), being a closed curve

surrounding the origin, it is shown that this set can be parametrized by

means of perfect Euler !-splines E(p2;h,•) corresponding to the operator

(D) (cf. Definition 3.2.12). For this purposesome elementary properties

of E(p2;h,•) are needed, which are contained in Subsectien 5.5.2. In Sub­

sectien 5.5.3 the main theorems for the full-line case are proved; it also

contains a few examples. Subsectien 5.5.4 is concerned with a parametriza-+ tion of arm(p2) in terms of the so-called one-sided perfect Euler !-splines

+ E (p2;h,•). In general, several cases have to be distinguished corresponding

to the various possibilities for the location of the zerosof p2 .

As remarked earlier our approach to deal with Landau problems is based on

investigating the sets ar (p) and ar+(p ). Consequently, the results of m n m n this sectien and subsequent sections have been obtained within that frame-

werk. We cbserve that for second order differential operators, these results

can be derived by different means.

5.5.2. Some elementary properties of E(p2;h,•)

We reeall that E(p2 ;h,•), the perfect Euler !-spline corresponding to the

operator p 2 (D), is uniquely determined by the conditions

i) E(p2;h, • l E c (1) (IR) I

(5. 5 .1) ii) E(p2 ;h,t+hl - E(p21 h,t) (t E JR) ,

iii) p 2 (D)E(p2 ;h,t) = - 1 (0 < t < h) ,

provided (-1} t o.

REMARK. If p 2 (-1) = 0 (cf. (3.2.1}) then a nontrivial function f E Ker(p2l

exists such that f(t+h) - f(t) (tE JR). However, this only occurs if p 2 has purely imaginary zeros Àl, 2 ~ ± iB (B > 0) and if h = (2k + 1) 11/B with

k E N0

•

114

In order to obtain some elementary properties of E{p2;h,•), it is of imper­

tanee to assume that p2 (D) is disconjugate on (O,h). The maximal lengthof

an interval of disconjugacy for the operator p2 (D) is equal to the distance

between two consecutive zeros of the corresponding fundamental function ~

(cf. p. 6). If p 2 has zeros Àl 2

=a± iS with 8 > 0 1 then ~(t) = -1 at ' ·

= 8 e sin(Stl and thus the maximal length is equal to 7r/B. :l;f the zeros

of p2 are real, then p2 (D} is disconjugate on m. Associated with the monic

* polynomial p2 we define the number h by

(5.5.2) {

7f

* s h := ..

(À1

= a+ iS , 8 > 0} ,

Consequently, every nontrivial function f E Ker(p2l has at most one zero in

[o,h*), counting multiplicities.

* It fellows from the remark above and the definition of h that for all

* * h < h the function f(p2;h,•) is well defined. Moreover, if h < .. then

* E(p2;h ,•} is well defined .in case the complexzerosof p2 have nonzero

real parts.

For the purpose of parametrizing the set armcp 2J we introduce the curve y

defined by

(5.5.3}

Because f(p 2;h,•) is periadie with period 2h (cf. (5.5.1)) it 1immediately

follows that y is a closed curve. The next lemma is used to deduce another

property of y.

LEMMA 5.5.1. Let (a,bl TE m2 !JJith (a,b) Y, (0,0) andlet h* bedefinedby (5.5.2).

If 0 < h < h* or ifh h* in case h* < oo and ; 2 (-1) # 0, then the funation

aECp2;h,•) + bE' (p2;h,•) has exaatly one zero in [O,hl~ aounting multiplia­

itiea.

PROOF. The conditions for h imply that ECp2;h,•) is well defined and that

the operator p2 (D) is disconjugate on (O,h). Let the function s be defined

by

(t E :R) •

By (5.5.1) onehasp2 (D)s'(t) =0 (tE (O,h)) ands(t+h) =-s(t) (tEm).

If s (0) # 0 then s (hls (0) = - s 2 (0) < 0 and thus s has an odd number of

115

zeros in (O,h), counting multiplicities. Therefore, if s has at least three

zeros in (O,h) then s' has at least two zeros in (O,h). However, the oper-

* ator p 2 (D) is disconjugate on (O,h) since h ~ h Hence s' vanishes identic-

ally on (O,h). This contradiets s(h) =- s(O) # 0. If s(O) =- s(h) = 0

then the assumption of at least one zero in (O,h) again leads to s'(t) 0

(te (O,h)}. Hence s(t) = 0 (t (O,h)), contradicting (a,b) :f (0,0).

Consequently, s has precisely one zero in [O,h) 1 counting multiplicities. 0

It fellows that 1 under the condition of the lemma, every half line through

the origin intersects y in precisely one point.

Now the aim is to de termine the number h such that 11 E (p2

; h, • l 11 m, where m

is a prescribed positive number. For this purpose Mp2 as defined by (5.3.3)

is needed. Recalling that Àl and À2

denote the zeros of the manie polynomial

Pzl one can easily verify that

(5. 5 .4) (\ = a+ i!3 , a. 'i 0 , !3 > 0) ,

(otherwise) •

The following lemma showshow IIE(p2

;h 1 •)11 depends on h. As the proof consists

of a straightforward computation 1 it is omitted here.

* LEMMA 5.5.2. Let p 2 e " 2 he a monia polynomial and let h and Mp 2 be given

by (5.5.2) and (5.5.4), T>espectively. Then the funotion h~IIE ;h,·JII is

oontinuous and stT'ictly inoreasing on (O,h*l and it has the properties:

i)

ii) if h *

iii) if h *

< 00

then lim IIE!p2

;h,•)ll h--

and Mp2

then < 00 lim hth*

< oo and Mp2 then = 00 lim

hth*

11 E (p 2 ; h, • l 11

11 E (p 2 ; h I • ) 11

Moreover, in all these caseslim IIE(p2 ;h,·JII 0. MO

116

* If h : ® and Mp2 < oo, i.e., if the zerosof p 2 arerealand p 2 (0) ~ 0,

then again a straightforward computation shows that

(h -+ ®) I

uniformly in t on ever.y bounded subinterval of lR, where E Cp2

; 00 , •) is the

unique bounded function in C ( 1 l ( lR) determined by

(5.5.5) (t € lR) •

It easily fellows that 11 E Cp2

;"', •) 11

5.5.3. The full-line case (J =lR)

In this subsectien we shall prove that the curve y, defined by {5.5.3) is

exactly the boundary armcp2) we are looking for. The two cases 0 < m < Mp2 and m = Mp2 are treated separately.

The following result is one of the main theorems of this chapter.

THEOREM 5.5.3. Let p 2 Cnl be a diffePentiaZ opePatol' of OPdel' wo and Zet m

be suah that 0 < m < Mp2

• Then there exiets a unique h ~ith 0 < h < h* suah

that

* PROOF. According to Lemma 5.5.2 there exists a unique h E (O,h ) such that T

IIECp2

;h,•)ll = m. Let (x,y) E arm(p2>. Then by Lemma 5.5.1 there exists a

positive a and a t 0 E [0,2hl such that x= aECp2;h,t0 l and y = aE' (p2;h,t0J.

We prove that a 1. As (x,y)T Er (p2) a function f E F (p2 ,JRJ exists m m ,

such that f(t0

J x and f' Ct0 l = y. Without loss of generality we may assume

that t 0 " [O,hl. Let g: f-aECp2 ;h,•), then g(t0

) = g'(t0J = 0 and thus

according to Taylor's formula (cf. Lemma 1.4.4)

(5.5.6)

t

g(t) = f cp(t-T)p2(P)g(T)dT I

to

where cp is the fundamental function corresponding to p 2 (n).

Since n 2 (n) is disconjugate on (O,h*J and 0 < h < h*, one has cp(t) > 0

(0 < t < hl and cp(t) < 0 (-h < t < 0). We shall show that the assumption

cr > 1 leads toa contradiction. As a consequence of (5.5.1) we have

p2

(D)g(t) = p2

(D)f(t) +cr > 0 (a.e.)

and thus in view of (5.5.6), g(t) > 0 (0 ~ t ~ h, t # t0). Hence

As by (5.1.2) l!fll <:; m, a contradiction is established if we prove that

E (n2;h, •) attains its maximu.m m at a point in [O,h]. 1\.ccornin<J to Le!'lma

5.5.1 there is exactly one point t1

E (O,h] such that E' (p 2;h,t1l = 0. In

view of (3.2.30)

( 5. 5. 7) ~i f --=---------- dz (0 :$ t s; hl .

c

117

By applying the residue theorem we can compute E' (p 2;h,t) explicitly. It

then follows from the sign structure of E' (p 2 ;h,•) that t(p2 ;h,•) attains

its maximum at a point in (O,h]. Consequently cr <:; 1. If cr < 1 then because

of the fact that

t~e point (x,v) is nota boundary ~int of rm(p2), contrary to the assump­

tion made. Hence cr = 1.

In order to

and h* = oe

the curve y

* treat the case m = Mp2 , we have to distinguish between h < oe

* If h <oe then it can be shown that the set 8fm(p2) consists of

(cf. (5.5.3)) with h = h*. The proof is completely similar to

* the proof of Theerem 5.5.3. If h oe, i.e., if the zerosof p 2 are real

and p 2 (0) # o, then the function E(p2 ;oc,•) may be used to describe arm(p 2l.

However, the curve {(E(p2

;oc,T),E' (p 2 ;oc,T)}T \-oe< T <oe} is not closed; as

can be shown by a simple computation, it only describes half of the set

arm(p2). By a similar reasoning as given in the proof of Theerem 5.5.3 one

ultimately arrives at the following

THEOREM 5. 5. 4. Let p2

(D} be a differential operator of order tl.Uo suah that

Mp2 < "'• Then arMp2

(p2) is given by

118

0 ~ T < 2h*}

* (h =co}

The contents of Theorems 5.5.3 and 5.5.4 are intimately connected with the

Landau problem for the operator p2 (D}, as has been pointed outinSection

5.4. Consequently, we have the following

COROLLARY 5.5.5. Let p 2 (o) be a diffe~ential ope~to~ of order two and let

0 < m < Mp2• Then the~ exists an h E (O,h*) suah that for a~bitrary (a,b) T E lR2

(5.5.8)

Moreover, if m

max llaf+bf'll fEFm (p2 ,JR)

* Mp2 and h < "' then (5.5.8) also holds fo~ h * h •

We proceed by discussing a few exaroples intended to illustrate the theorems

just given. In each case a complete description of the set rm(p2) is ob­

tained.

EXAMPLE 1. p2(D) 02.

As

2 - ! t2 +~t (0 ~ t ~ h) E(D ;h,t) 2 2

the curve y (cf. (5.5.3)) is given by the equation

2 Hence, according to Theorem 5.5.3 with m = h /8,

As an immediate consequence it follows that

(5.5.9) lf' (tl! ~ hcm- ifftl I>

,

which implies Landau's well-known inequality llf'll ~ l2ii (cf. Theerem

5.1.1).

EXAMPLE 2. p 2 (0) ~ o2

I.

In view of (5.5.4) it fellows that Mp2

~ 1. Since

2 E (0 -I:h,t) 1 _ cosh (t- h/2)

cosh (h/2)

the curve y is given by the equation

2 2 -2 y -x +2lxl = (cosh(h/2))

Hence, according to Theorem 5.5.3 with m =

ar <o2- rl

T I 2 2 + 2lxi (5.5.10) = { (x,y) y -x

m

It can be shown, using Theorem 545.4 with h *

(5.5.11) ar 1 <o 2- Il ~ {(x,y) T I !x + lyl 1}

(0 <;; t <;; h) 1

-1 cosh (h/2),

2 = 2m-m, lxl s

oe, that if m ~

m}

1 then

A sketch of the set r (o 2 - I) for m = 1 and m < 1 is given in Figure 1. m

V

-1

x

·1

Figure 1.

As a consequence of (5.5.11) the inequality lf(t} I + lf' (t) I <;; 1 (tE~)

holds for every bounded f E AC (2) (IR) satisfying 11 f" - fll <;; 1. Schoenberg 2

[53] obtained the best upper bound for l!f' 11 on Fm (D -I, lR). Sharma and

Tzimbalario [58] observed that the upper bound obtained by Schoenberg is

also valid for the functional llf' -afll in case lal< (m(2-m))~. This

119

120

assertien is also easily inferred from Figure 1, since the restrietion on a

agrees with the value of the two tangents at (O,(m(2-m))~).

5.5.4. The half-line case (J =E~)

The role of the perfect Euler !-splines for the full-line casel is now taken

over by the so-called one-sided perfect Euler l-splines. For tpe polynomial

case these functions are introduced by Schoenberg and Cavaretta [54] as

limits of sequences of special spline functions having knots in E~. With

respect to the operators n2 and n3 the one-sided perfect Euler splines are

simply related to the perfect Euler splines. In this subsectien we only

need the one-sided perfect Euler l-spline corresponding to an arbitrary

differential operator of order two. Therefore this function is introduced

in relation to its corresponding perfect Euler t-spline.

Let f(p2;h,•) be the perfect Euler l-spline corresponding to the operator

p 2 (D) and let 0 < h < h*, where h* is given by (5.5.2). We consider the

t-spline s, corresponding to the operator Dp2 (D), with knots h,2h, ••• and

coinciding with f(p2;h,•) on E~. This amounts to removing the knots

O,-h,-2h, •••• The function E'(p2 ;h,·} has exactly one zero in (O,h] where

f(p 2;h,•) attains its maximum m (cf. the proef of Theerem 5.5.3). Now the

largest value t0

of t is determined such that t0

s 0 and s (t0

) = - m; for

the moment we take its existence for granted. Then a one-sided perfeat

Euler t-spline, denoted by E+(p2;h,•), is defined by

(t ~ 0) •

We cbserve that the perfect Euler t-spline and its one-sided version coin­

cide for t ~ -t0 •

With respect to the existence of t 0 we remark that this is guaranteed in

case p2 has real zeros, as then s' has only one zero on <-~,hl and

limt+-ro s(t) = -Mp2 < -m.

If p2 has nonreal zeros an explicit computation is needed to prove the

existence of t 0 • By way of example this will be shown for the operator 2 p 2 (D) = (D +ai) +I, with a > o.

At some value t 1 E (O,h] with (cf. (5.5.2)) h < h* = n, the function s

attains its maximum m. Since p 2 (D)s(t) = -1 (t < h), s(t1) = m, and

s'(t1) = 0, it fellows that

Hence

1 ( 1 ) -a (t-t1) s(t) --2-- + m + -

2-- e (cos(t-t1) +a sin(t-t

1)) •

a+l a+l'

--2---(l +1

( m + -+-) a + 1'

(l7f e < - m.

This relation ensures that t0

E (t1-7T,O].

The following theorem emphasizes the importance of the one-sided perfect

Euler L-splines with respect to the Landau problem on the half line.

THEOREM 5.5.6. Let p 2 be a monia poZynomiaZ and Zet m be suah that * 0 < m < Mp 2• Then there exists an h E (O,h ) suah that for eaah ~ E ~

max + 11 ~f + f '11 + , fEFm (p2,JRO)

121

where Fm(p2 ,JR~) is given by (5.1.2) and E+(p2;h,•) is the one-sided perfect

EuZer t-spZine defined by (5.5.12).

* PROOF. Because of Lemma 5.5.2 there exists an h E {O,h ) such that

llf(p2

;h,•)ll = m. Taking into account the construction of f+(p 2 ;h,•) on

p. 120 we conclude that E+(p2

;h,O) -m and that E+(p2

;h,•) increases

monotonically from - m to i ts first maximum m at t = t 1 , say. Let qJ be the

fundamental function corresponding to the operator p 2 (D) and let

qJl E Ker(p2l be the function satisfying qJl (0)

T E [O,t1J is determined in such a way that

(5.5.13) qll (tl- 'r)

1, qJl (0) = 0. Now

If such a -c does not exist, then -c is taken to be zero. We assert that

In order to prove (5.5.14) we apply Taylor's formula (1.4.4) with n 2,

a = -c, and g E Ker(p2) given by

g{t) = f{-c)tpl(t--c) + f'("C)tp(t-"C) {t E ~) •

Consequently,

122

t1

(5.5,15) f(tl) = f(T) cpl (t1 - T) + f' (T) cp(tl- T) + J cp(tl- ')Pz (D) f(l;)dt ,

'[

If T ~ [O,t1J satisfies (5.5.13), then (5.5.15) can be rewritten as

' tl

(5.5.16) cp(t1 --r) (f' (T) +llf(T)) = f(tl) - f cp(t1 - ~)p2 (D)f{,)d~ •

'[

* Since [O,t1J c [O,h l the operator p 2 (D) is disconjugate on [O~t1 J and thus + cp(t1 - ') > 0 (0 :S ' < t 1). Assuming f ~ Fm(p2 ,JR0), we conclude from

(5.5.16) that

(5.5.17) I \lf {T) + f' {T) I

Substituting the function f+(p2;h,•) for f into {5.5.16) we obtain

Hence {5.5.17) implies that

lllf(T) +f' (T) I $ \lf+(p2;h,T) + f~(p2;h,T)

and (5.5,14) easily fellows.

Now suppose that no T ~ [O,t1J exists satisfying (5.5.13). Then

Since cp{t1l > 0 there is a positive v such that w1 (t1l- (JJ +v)cp(t1l = 0.

In view of this one arrives, by substituting T = 0 into (5.5.15), at

tl

cp(t1HJJf(O) +f' (0)) .. -vcp<t1

)f(O) +f(t1>- J cp(t

1 -!;)p2 {D)f(~ldt •

0

Consequently,

t1

IJJf(Ol +f' (0) I :S cp(!11

(vcp(t1)m + m + f cp(t1 -E:ldt)

0

123

and again (5.5.14) is established. This completely proves the theorem. 0

-m

I I

I

2 D -I and 0 < m < Mp2

, ,.

y

l --- ( m(2-m))2

m x

Figure 2.

1 1 is given below.

+ 2 2 ' In order to oompare the sets r m (D - I) and r (D - I) Figure 2 a lso contains

2 m 2 + 2 the set r (D I) (cf. Figure 1 1 p. 119). Obviously 1 fm(D -I) c:. rm(D -I).

m + 2 Note that the part of arm (D -I) for which x > -m and y > 0 may be para-

metrized by (f(p2

;hiT) ~E' (p2

;h,T)) and that the boundaries of rm(o2

- I) and

r+(D2 -I) coincide for {(xly}T I x~ OI 0 ~ y ~ (m(2-m))~} and for m T

{(x,y) I x~ 0, -(m(2-m))~ ~ y.::; 0}.

Having dealt with 0 < m < Mp2

, we now consider the case when m ~ Mp2 . If

the zeros Àl and À2

of p 2 have positive real parts then 1 according to + Lemma 5.3.7, we have rm(p

2l = rm(p2 l~ and the results of Theerem 5.5.4

apply. With respect to the location of Àl and À2

two cases remain to be

considered, viz. Àl < 0, À2

> 0, and Re(À1

) < 01 Re(À2

) < 0.

THEOREM 5.5.7. Let the ze ros À1 and À2 of the monic polynomiaZ p 2 satisfy

À1 < 0~ À2 > 0 and Zet Mp2 <;; m < "'~ where Mp2 is given by (5.5.4). Th en

+ r m (p2) { (xly)

T I lxl <;; m, ly- À1xi s -1}

À2

124

-1 PROOF. Let m ~ Mp2 = (-À1

À2

) (cf. (5,5,4)),

I I -1 + f' (0)- À 1 f(O~ ::; À

2 for all f E: Fm(p2 ,JR

0l.

bounded on E 0 and satisfies the differential

We first show that

The function g := f' - Àl f is

equation g' - À2g = p

2 (D) f.

This equation has a unique bounded solution which is easily seen to be

I À

2 (t-T)

g(t) = e p2 (D)f(T)dT (t ~ 0) •

t

Hence (cf. (5.1.2))

I g(O) I

It fellows that

-1 Now let (x,y) satisfy y- À1x = À2 and I x I ::; m. We must prove that

(x,y)T € f:(p2l. If x= (-À1

À2)-1 and y = 0 it is sufficient to cbserve -1 +

that the constant f~~ction (-À 1À2) belengs to Fm(p2,JR0l.

I f -m ::; x < (-À 1

À 2

) then the function f 1 gi ven by

+ T + is an element of Fm(p2,JR0) and therefore (f1 (t),fl{t)) € rm!p2l (t ~ 0).

Now -m s f 1 (t) < (-À 1À2)-1 and fi (t) -À 1f 1 (t) =À;1•

If (-À À )-1 < x ::; m then the function f 2 given by 1 2

T + can be used to show that (x,y) € rm(p2).

A sketch of r: (p2l , where p 2 (D} = o2 - I and m ~ 1, is gi ven below.

0

x

Figure 3.

The only remaining case is Re(À 1) < 0, Re(À2

) < 0, and m Mp2 . Let g be

the function satisfying p2 (D)g(t) = 1 (t > 0), g(O) mand g' (0) = 0.

Hence, if À1 ,2

=a± iS with a < 0 and S > 0 then

2 2 -1 2 2 -1 at -1 g(t) = (a +13 ) + (m- (a +B ) )e (cos(St}- (ai3} sin(i3t)} •

* The function g has a maximum at t = 0 and for some value t < 0 it in-

creases from -m to m on [ ,0]. The function pm(p2 ;•) is now defined as

(0 :; t < t*) ,

Evidently, pm(p2;·) E Fm(p2 ,JR~). Moreover, pm(p2 ;•} has the following

extrema! property, the proof of which is omitted as it is similar to the

proof of Theorem 5.5.7.

THEOREM 5.5.8. Let p 2 e be a monic polynomial whose zePos À1 and À2 satisfy Re(À 1} < 0, Re(À 2) < 0 and let m ~ Mp2, whePe Mp2 is given by

(5.5.4). Then for eaah ~ E ~ one has

11 ~p (p •• ) + p ' (p •• ) 11 m 2' m 2' + max + ll11f + f'll +

fEFm(p2,JRO)

125

126

5.6.1. Introductory ramarks

In the previous sectien we discuseed in detail the Landau problem for

general second order differential operators in the full-line case and in

the half-line case. For general third order differential operators the ana­

logous problems are much more complicated. Bowever, a signific4nt simplifi-* . cation occurs if the operators have the proparty p3 (D) = p 3 (D}; this w~ll

be assumed throughout this section. This introduces a certain symmetry

which makes the problem easier to handle. Consequently, only operators of

the form p3 (D) = o3 +cD (c E :R) will be considered. Furthermore, we only

consider the case B2 := c > 0 ((3 > 0); the analysis and the results for the

other two cases c < 0 and c 0 are quite similar. We note that results for

the case c = 0 may be found in Ter Morsche [44].

In Subsectien 5.6.2 it is shown that arm(p3) can be parametrized by means

of the perfect Euler !-splines c(p3;h,•). According to Definition 3.2.12

~e function f(p3;h,•) is defined for those values of h for which

P 3 (-1) "# 0. Since (cf. (3.2.1))

(!i!: E t:l ,

one has p3 (-1) = -4(1+cos(Bh)) and therefore the restrietion h '# (2k+1)71/(3

(k € .1':) must be made. It easily follows from (3.2.30) that foi1 0 $ t $ h <

< 71/(3

(5.6.1)

We note that the fundamental function ~ corresponding to the operator

o3 +B2D is given by (cf. Definition 1.3.3) ~(t) = (3- 2(1-cos Bt) and there­

fore (cf. Lemma 1.4.3} o3 +e2 o is disconjugate on (0,271/8). Since

E' (p3;h,•} is an Euler .C-spline corresponding to the operator o3 + 82 0, it

follows from Lemma 3.2.8 that the function E' (p3;h,•) has precisely one

!i!:ero in [O,h), provided 0 < h < 71/B.

An easy computation shows that

(5.6.2) llf(p3;h,•lll =- f(p3;h,O) = f(p3;h,h)

B-3 (tan(68h) - lBh> (0 $. h < .!..) (3

127

In what fellows the abbreviation M(h) := (tan(Sh/2) - Sh/2) will be

used.

5.6.2. The full-line case (J = ~)

We reeall (cf. Theorem 5.5.3) that the curve consisting of the points

(f(p2;h,T),f' (p2 ;h,T))T (0 $ T < 2h) is 3rm(p2), where 0 < m < Mp

2. Now it

will be shown that a similar phenomenon occurs for the third order operator 3 2

(D) D +S D. However, the function f(p 3 ;h0 ,·) with h0

E (0,11/Sl satis-

fying llf(p3 ;h0 ,·lll = m (cf. (5.6.2)) generates a closed curve on the two­

dimensional surface arm(p3). Therefore we have to look for other extremal

functions. It turns out that one also obtains extremal functions if appro­

priate constants are added to the perfect Euler !-splines f(p3;h,•), where

0 s h $ h0 • Details are gi ven in the following

THEOREM 5.6.1. Let m > O, let f(p3;h,•) be the perfect Euler t-spline oor-

responding to the operator p3

(D) D3 + S2

D 1.Jith S > 0 and, furthermore,

Zet h0 E (0,11/Bl be such that IIE<p3;h0 ,·lll M(h0 l m. Then arm(p3 l aan be

parametrized by

(5.6.3)

2h, (J ± 1} •

PROOF. Let the function f be defined by

(5.6.4) f(t) := f(p3;h,t) + cr(m-M(h)) (tE~) ,

We first show that (f(T) ,f' (T) ,f" (T)) T E arm (p3) if 0 s h s ho, 0 sT s 2h,

and cr E {1,-1}. If h = 0 then f(p3;h,t) 0 (tE~) and M(h) = 0, and thus

we obtain the two points ± (m,O,O)T which obviously belong to arm(p3). If

0 < h s h0 , then 0 < M(h) s m. Consequently, in view of the definition of f

and (5.6.2) we get

{ :M(h)- m'

(cr 1) , max f(t} f(h)

Osts2h m (cr -1) , (5.6.5)

{: m 2M(h) ~ -m (cr 1) ,

min f(t) f(O) OstS2h (cr -1)

128

Hence !lfll =mand llp3 (~)fll = llp3 (n)f(p3 ;h 1•)1J = 1. Therefore f € FmCp31lRl

and (f(T) ,f' (T) ,f" (T)) . € f m(p3). In order to prove that

(f(T) 1f'(T) 1f"(T))T € ar Cp3

) we assume that T <:: [O,h] 1 since a similar . m T

argument may he used if T <:: (h,2h]. If (f(T)~f'(T),f"(T)) i armcp3

) then

because of the convexity of rm(p3) and the fact that Q is an interior point

of r (p3l (cf, Theorem 5.3.3) 1 there exists a number À > 1 such that m T

À (f (t) 1 f' (T) 1 f" (t)) E ar (p3) • Let q € Fm (p3

,lR) be a functie~ satisfying (i) (i) m

q (t) = Àf (t) (i= 0,1 12), then the function r given by

r(t) := q(t)- Àf(t) (t E JR)

satisfies the relations r(i) (T) 0 (i = 0,1,2) and

for almest every tE [O,h]. Using Taylor's formula {cf. Lemma 1.4.4) 1 we

obtain

(5.6.6) r{t) r Ql(t- ~, (p3(D)q(~) +À)d~ T

(0 s t s h) •

-2 Here Ql ( tl = 8 ( 1 - cos (St)) , the fundamental function corresponding to the 3 2

operator D + 8 D. It follows from (5.6.6) that r{O) ::> 0 and r(h) ~ 0.

Hence q(h) ~ Àf(h) and q(O) s Àf{O). As a consequence of {5.6.5) we conclude

q(h) ~ Àm > m or q(O) S -Àm < -m, contradicting q € F (p31lRl. This centra-T m

diction es tablisheS that (f (T) 1 f 1 (t) 1 f" (T)) € ar (p3

) , T m

We preeeed by showing that for all (x 1y,z) € 3rm(p3 ) there exist numbers * * * * h E [O,h

0], t E [0,2h ] 1 and a E {-1,1} such that (x,y,z)

= {f(t*l, f' (T*l ,f" <•*>). If y = z = 0 then x = ± m, and we may take

h* = ,.* 0, a* = ± 1. It remains to consider the case (y,z) .; (0,0). We

note that if g <:: F (p3

,lR) then the functions - g and tr-:tg(-t) also m T

belong to F (p3

,lR). Consequently, if (x,y,z) € r (p3

) then also Tm T m

- (x,y,z) € fm(p 3l and (x,-y,z) E rm(p3). Because of this symmetry we may

* ~ 0 and z s 0. It will now be shown that numbers h € (0 ,h0 J,

a* = ± 1 exist such that f(T*l = x, f' <•*> = y, and f" <•*> = z

assume that y

,* E [O,Ih*J, T

if (x,y,z) E arm(p3). Using (5.6.1) and (5.6.4) we obtain the following

equations for h*, ,.*, a*:

i)

(5.6.7) i i)

* * * * ECp3

;h ,T) + cr (m M(h ))

cos(S(T*- !h*l)

cos<! (:lh *>

2 13 y + 1

X I

Given y ~ 0 and z ~ 0 with (y1zl ~ (0,0) it easily follows that equations

i) and ii) of (5.6.7) uniquely determine h* E (0,~/B) and ,* E [Oih*/2].

Having determined these numbers h* and T* our next purpose is to find

129

* * * * * o E {-1,1} such that Ecp3 ;h 1T} + cr (m-M(h )) x. Let m1

be the smallest

constant satisfying

(5.6.8) ii) = x

or

If m1 is determined by the first equation of ii} of (5.6.8}

* else o -1.

* then a

Note that m1

= m has to be proved. Let h1

E (0 1 ~/(3) satisfy M(h1

) = m1

• We

conclude that (X 1 Y 1 Z) = (f(T*l 1f' (t*l 1f"(T*)) 1 where fis given by (5.6.4)

* * T with m replaced by ml, h by h I and cr by a Consequentlyl (xlylz) E arml (p3)

by the first part of the proof. Since 1 by a simple reasoning1 arm1

Cp 3J c

c: r;2

(p3) (m2 > m1J (cf. p. 2 for notation) 1 and because of the assump­

tion (x,y,z)T E arm(p3)1 we infer that m ml. Therefore hl ho and

* 0 < h s h0

< ~/(3. This completely proves the theorem. 0

REMARK 1. Theerem 5.6.1 implies the following assertion. If W is a tunetion

defined on ~3 such that Iw I attains its maximum on rmcp3) at arm(p3), then

the functional

fHI!W(f,f' ,f"lll := sup jw(f(t) ,f' (t) ,f"(t) I tElR

attains its maximum for a perfect Euler f. -spline plus an appropriate con­

stant function. In particular, this will be the case when W is a linear

function. This gives rise to the following

130

COROLLARY 5.6.2. Let m > 0~ let E<p3;h0 ,•l be the perfect Euler l-epline

corresponding to p3

(D) = o3 + a2 D with B > 0 and .. furthermore, let

h 0 E (0,1T/Bl be swh that IIE{p3;h

0,·JII =m. Let p

2 E1T

2 be an arbitrary poly­

nomial. Then there ei:ists an hE [O,h0

J and a constant function d such that

max !lp2 {D) fll = llp2 (D) (E(p3 ;h, •) + d) 11 •

f€Fm(p3,JR)

3 REMARK 2. The corresponding results for the operator p 3 (D) = D , which may

also be found in Ter Morsche [44], can easily be obtained by letting B tend

to zero. A sketch of rmco3J is given in Figure 4. We cbserve that rm(D3

l is

not strictly convex since its boundary contains straight line segments (for

instance, the line segment PQ) •

z

2 A {o,t{3m)"!",o)

y

x

Figure 4.

131

According to Remark 2 a parametrization of 3f (DJ) can be derived as fol-3 m

lows. Letting S tend to zero in M(h) := S- (tan(~Sh)- !Sh) (cf. p. 127) we

obtain M(h) h3/24. Since M(h0) = m (cf. Theorem 5.6.1) it follows that

h0

= 2 (3m) 113 • Moreover, the perfect Euler spline E (D3; h,•) is obtained

from (5.6.1) by letting S tend to zero. In view of Theorem 5.6.1 we then

have

{ E 3 1 3 E, 3 E" 3 T ( (D ;h,T) +a(m-24

h ), (D ;h,T), (D ;h,T))

0 s h s 2 (3m) 113 , 0 s T s 2h, a ± 1} ,

where

3 f(D ;h,t) (0 s t s h) •

On the boundary of r (D3) we have drawn two closed curves denoted by 1 and m

2, respectively. Curve 1 corresponds to the perfect Euler spline E(o3 ;h,•) . 1/3 3 1 3 wJ.th h 2(3m) • Curve 2 corresponds to E(D ;h,•) + (m-

24h) with

h = (12m) 113; it obviously contains the point B = (O,O,!hl.

With respect to the Landau problem it is evident from Figure 4 that

llf'll s !(3m) 2/ 3 and llf"ll s (3ml 113 whenever llf'"ll $1 and llfll s m.

132

6. ON TfiE LANVAU PROBLEM FOR PERIOVIC FUNCTIONS

6 . 1. I n:tlto du.c:ti.o n. a.n.d ~ umma~~.y

In Sectien 5.1 of the preceding chapter the Landau problem has been defined

as the problem of determining the best possible upper bound for a functional

of the form ft--tllpk(D)fiiJon Fm(pn,J). We reeall (cf. Sectien 5.4) that for + J =~ or J =~0 this problem is equivalent to determining the maximum of a

linear function on the convex sets r (p ) and r+(p ) , respectively. m n m n In this chapter we shall deal with the Landau problem for periodic functions.

In order to obtain an appropriate setting for the problems involved, a few

definitions will be given first. In these definitions, and also throughout

this chapter, T is a positive number denoting the period of the periodic

funetions. We reeall (cf. p. 75) that periodie functions with period T are

called T-periodic.

DEFINITION 6.1.1. The sets W(n) (lR,T) and L00

(lR,T) are defined by

w<n> (lR,T) := {f E w<n> (lR) I f(t+T) = f(t), tE~} ,

L (~,T) := {f E L {lR) I f(t +T) = f(t) (a.e.), t E :R} , 00 ""

!J)he:r>e w(n) (~) and L00

(lR) a:r>e given on p. 3 of Subeeation 1.2.3.

DEFINITION 6.1.2. Given a diffe:r>ential ope:r>ato:r> pn(D)~ the set F{pn,T) is

defined by

F(pn,T) := {f E w(n) (lR,T) I IIPn (Dl fll :;; 1} •

We preeeed by defining the set f(pn,T), which is related to F(pn,Tl in the

same way as fm(pn,lR,O) is related to Fm(pn 1 lR) (cf. {5.1.2) and (5.1.3}).

133

DEFINITION 6.1.3.

(6.1.1) { , (n-1) T I } (f(O),f (O), ••• ,f (0)) f E F(pn,T) .

Now the general Landau problem for T-periodic functions can be defined as

the problem of determining the best possible upper bound for a functional of

the form f~llpk(D)fll on F(pn,T); again (cf. Sectien 5.4) this problem is

equivalent to determining the maximum of a linear function on the set

f(pn 1 T). However, in contrast to fm(pn) (cf. Theerem 5.3.3), the set f(pn 1 T)

may be unbounded; this occurs when Ker(pn) contains nontrivial T-periodic

functions. In conneetion with this we define the set Ker(pn,T) as fellows.

DEFINITION 6.1.4.

f (t) 1 t E JR} •

The set Ker(pn,T) is spanned by the functions tl-'1' sin(2kTrt/T),

t~--+ cos (2k1Tt/T), where k varies over the set of integers for which

pn(2kTri/T) = 0. Consequently, as pn has real coefficients it fellows that

(6.1.2) Ker(pn,T) = Ker(p~ 1 T)

Tne main contents of this chapter are as fellows. Sectien 6.2 investigates

conditions on u E L00

(JR,T) for the differential equation pn(D)f u to have

a salutionfin w<nl (JR,T). The uniqueness of such a salution is ensured by

restricting f to a set F(pn,T) c F(pn,T); this is done in Lemma 6.2.2.

Corresponding to F (p ,T) we' then de fine r (p ,T) analegeus to r(p ,T) in (6 .1.1) . n n n

In Sectien 6.3 we study the extrema! functions for the Landau problem with

respect .to periadie functions. In this conneetion a few properties of the

sets F(pn,T) and r(pn,T) are derived. For instance, it is shown that

r.(pn,T) is a strictly convex and compact set having Q as an interior point.

The subject of Sectien 6.4 is connected with werk done by Northcott [46] on

the problem of maximizing IIDkfll (0 ~ k ~ n-1) on F(pn,T) • Our main result

states that perfect Euler L-splines are extremal functions for the linear

functional f~ 11 (apk (D) + bpk+1 (D)) fl!, f E F (pn ,T) , under the following

conditions: i) pn has only real zeros with pn(O) 0; ii) the menie poly­

nomial pk+l is a divisor of pn with the additional property

134

Pk+1 (t) = (t-a)pk(t). Finally, Section 6.5 contains a parametrization of - 3 the boundary of the specific set f{D ,1).

6.2. A áw pJteU.minaJty lemma.; and the. &et.6 F(p ,T) and r (p ,T) n n

As remarked earlier, the set r{pn,T) is not necessarily bounded. ~e main

purpose of this sectionis to introduce a specific bounded subset f(p ,T) of n

f(pn,T), which coincides with f(p0

,T) if this set is bounded itself. In

order to achieve this we start out by characterizing the functions

u € L00

(lR,T} for which the differential equation

(6.2.1) p0

(D)f(t) = u(t) (a.e.) (t € lR)

has a salution f € w (n) (lR, T) •

We note that f € W(n) (lR,T) implies that f is a T-periodic function with

the property

(6.2.2) f (1} (0} f (i) (T) (i 0,1, ••• ,n-1) •

LEMMA 6.2.1. The differentiaZ equation (6.2.1) has a soZution f € w<n> (lR,T)

if and onZy if

(6. 2. 3) J g(-r)u(t)d-r

0

0

PROOF.·Let f € w<nl (lR,T) :atisfy (6.2.1) and let g € Ker(pn,T). As (cf.

(6.1.2)) Ker(p0

,T) Ker{pn 1 T}, integration by parts yields

T T

J g(t)p0

(D)f(t)d-r = (-1) 0 I * f(-r)p0

(0)g(T)d-r .. 0 •

0 0

He nee (6.2.3) is necessary. To prove that (6.2.3) is sufficient we write

(6.2.1) as a system of first order linear differential equations hy putting

x1

:= f, x2

:"' f', ••• , x0

:= f(n-1). We then obtain

(6.2.4) x' = Ax + bu ,

T T where ~ = (x1,x2 , ••• ,x0

) , b = (0,0, ••• ,0,1) and where A is the correspond-

ing campanion matrix.

It is well known that a salution of (6.2.4) can be written in the form

(6.2.5) ~(t) etA ~(0) + r e(t-T)A bu(T)dT

0

135

We have to prove that fora given u E L00

(lR,T) satisfying (6.2.3), x(O) can

be chosen in such a way that x(T) ~(0), i.e. such that

(6.2 .6) I e(T-T)A bu(T)dT

0

Consequently, itmust be shown that 0

/Te(T-T)A ~u(T)dT is in the range of

(I- eTA). This is equivalent to the assertien that

(6.2. 7) y_T I e(T-T)A bu(T)dT

0

0

n T TA for all y_ E :R satisfying y_ (I - e ) QT; here the well-known fact is used

that the range of a linear transformation is orthogonal to the kernel of

i ts transpose. However, if y_ E :Rn satisfies y_T (I - e TA) QT then the func-. . T (T-t)A * hon g dehned by g(t) := y_ e ~ belengs to Ker(pn,T) = Ker(pn,T). So

(6.2.7) fellows from (6.2.3) and the lemma is proved. 0

We note that a generalization of Lemma 6.2.1 is contained in Brockett

[7, pp. 50-52].

For convenience we define

(6. 2. 8) {u E L", (lR,T)

T

f g(T)U(T)dT

0

0, g E Ker(pn 1 T)} .

We note that f E W(n) (lR,T) implies that pn(D)f belengs to Upn.

If pn(D)f =u E Upn then also pn(D) (f+g) =u for all gE Ker(pn,T).

Therefore, in order to ensure a unique salution of (6.2.1) we impose

additional conditions on the function f. This is done in the following

lemma.

136

LEMMA 6 • 2 • 2 • Gi Ven an aroi trary u E: Up ~ the di fferoentia l equation ( 6. 2 ; 1) n

has e:matly one solution f E: w(n) (lR,T) having the p:roperoty

(6.2.9)

T

J g('r)f(t)dr

0

0

PROOF. It follows from Lemma 6.2.1 that a solution f 0 of (6.2.1) exists.

Consequently, any solution f of (6. 2 .1} can be wri tten in the form f = f 0 + h,

where hE: Ker{pn,T). According to (6.2.9) we have to choose the function h

such that

T T

I g{t)h(t)dt I g(rl f 0 <r> dr (g € Ker(pn,T))

0 0

This uniquely determines h.

Lemma 6.2.2 motivates the introduetion of the following subset of F(pn,T).

DEFINITION 6.2.3. The set F(pn,T) is defined by

F(pn,T) := {f E F(pn,T) I J 0

g{T) f(t) dt

If Ker{p ,T) contains only the null function then F(p ,T) = F(p ,T). n _ n n Similar to r(pn,T) in Definition 6.1.3, the set r(pn,T) associated with

F(p ,T) is defined as follows. n

DEFINITION 6.2.4.

- (n-1) T I -r(pn 1 T) :={(f(O),f'{O), ••• ,f {0)) fE:F(pn,T)}

Properties of the sets F(pn 1 T) and r(pn 1T) that are of importance for the

Landau problem will be given in the following section.

0

137

The main result of this sectien is that f(pn 1 T) is a strictly convex and

compact subset of ~n having 0 as an interior point. Prelirninary to this it

will be shown that the elernen~s of W(n) (m 1 T) satisfying (6.2.9) can be

represented as convolution integrals.

LEMMA 6.3.1. If f E w(n) (m~T) satisfies (6.2.9) then

(6. 3 .1) f(t) = ~ r p (t-T)p (D)f(T)dT n n

(t E ~) 1

0

where P is a T-periodic function, the Fourier series ofwhich is given by n

(6.3.2) p (t) n

2k11it/T

k) p e (2k1Ti/T) Kpn n

(tE~) 1

Kpn being thesetof integers k for which pn(2k1Ti/T) = 0.

PROOF. It fellows frorn (6.3.2) that p(n-2) is continuous on n

0 (gE Ker(pn 1 T)) 1

~. Moreover 1

since Ker(pn 1 T) is spanned by cos(2k1Tt/T) 1 sin(2k1Tt/T) with k E Kpn.

Substituting (6.3.2) in the right-hand side of (6.3.1) 1 changing the order

of summatien and integration 1 and integrating by parts we obtain

2k1Tit/T IT t e e-2k1TiT/T pn(D)f(T)dT

T ki~p pn(2k1Ti/T) n 0

f(T)p (-D)e-2k1TiT/T dT n

f(t) (tE~) 1

( f (6 2 9)) _Tl OJT f(T)e-2k1TiT/T dT as c • • • 0 for all k E Kpn • 0

138

It w!ll be shown on p. 139 (cf. Remark 1) that if Kp = ' then P is an ex-n n ponential l-spline corresponding to the operator p (D), i.e., P € S{p ,T).

n n n However, P is an l-spline having double knots if Kp = {O} (cf. Remark 2,

n n p. 139) •

LEMMA 6.3.2. Let Pn be the funetion definèd by (6.3.2), Then Pn;satis~es

the differentia"L equation

(6.3.3) p (D)P (t) =- I e2k~it/T n n kEKp

(0 < t < T) I

n

where an empty sum is de~ned to be zero.

Furthermore~ ifQn denotes the anaZytie fUnetion coinaiding with Pn on (O,T)

then

(6.3.4) (t € lR) ,

where '!' is the fundamental funation aorr>esponding to pn (D) and c is given by

c = p(n-1) (T -) _ p(n-1) (O +) , n n

PROOF. A simple computation shows that.the residue of the function

(6.3.5) Tetz

z~ Tz (e - 1)pn (z)

(t € lR)

2k'lfit/T/ at the point z = 2k'lfi/T (k e ~. k { Kpn) equals e pn(2k'lfi/T).

It can be shown that the sum of all residues of (6,3.5) equals zero, pro­

vided 0 S t < T. As Qn coincides with Pn on (O,T) it follows from the

residue theorem and (6.3.2) that

(6.3.6) T f etz Q (t) = -~ dz ,

n 'lfJ. (eTz- l)p (z) c n

where c is a contour in the complex plane surrounding the zeros of pn and

exciuding the points z = 2k'lfi/T (k { Kpn) •

Consequently,

f tz

e dz (eTz- 1)

c

and so (6.3.3) follows by the residue theorem.

139

Since Pn E C {n-2) {lR) and Pn is

(j 0,1, ••• ,n-2) and therefore

T-periodic it fellows that p(j) (0) = P(j) (T) (j) (j) . n n

Q (0) Q (T) (J = 0,1, ... ,n-2). Let the

function f be given by n n ( .)

f(t) := Q (t+T) - Q (t) (tE lR). Then f J (0) = 0 n n

(j = 0, 1, .. , ,n-2) and

since the right-hand side of (6.3.3) is T-periodic. In view of the defini-

tion of <p (cf. p. 6) we then have f c<p for some constant c E JR. Hence

Qn (t +Tl - Qn (t) = cq~ (t) and the constant c equals

c = Q(n-1) (T) _ Q(n-1) (O) n n

p<n-1} (T -) _ p<n-1) (O +) n n

This proves the lemma.

REMARK 1. We cbserve that, if Kpn 0' then formula (6.3.6) agrees with

formula (3.2.9), with À and h replaced by 1 and T, up toa multiplicative

constant. Consequently, Qn(t) a~ 1 (pn;h,t) and Pn(t) = a~ 1 (pn;h,t) (t ElR)

for some constant a E JR. Hence P is an exponential !-spline corresponding n

to the operator pn(D) and with mesh distance T.

D

REMARK 2. If Kpn = {O} then, according to (6.3.3), the function Pn satisfies

the differential equation p (D) P (t) = - 1 (0 < t < T) • Since P E C (n-2) (lR) n n n

and Dp (D)P (t) = 0 on each subinterval (vT,VT+T) (v E~), the function P n n n

is an !-spline corresponding to the operator Dpn(D) having double knots at

the points vT (v E i?:) •

REMARK 3. If pn (D) = Dn and T 1 then - n! Qn is the ordinary Bernoulli

polynomial of degree n (cf. Nörlund [45], p. 65).

The next three results concern properties of the set f(pn,T) as introduced

in Definition 6.2.4.

THEOREM 6.3.3. The set r(pn,T) is a aonve:.c and aampact subset of lRn having

Q as an interior> point.

PROOF. Obviously, f(pn,T) is a convex set, since F(pn,T) is convex. With

respect to the compactness property we preeeed as fellows. In view of

(6.3.1} every f E F(pn,T} satisfies (cf. Definition 6.2.3)

140

Bence

T

f(t) = ~ J 0

p (t-T)p (D)f(T)dT n n (t € JR) •

T

:$ T JlPn(t-T)jd,

0

T

=~ fiPn(T)!dT

0

and so rep ,T) ç r (p) with m = T- 1 orTIPn(T)IdT (cf. {5.3.2)). n m n Since fm(pn) is a compact set (cf. Theerem 5.3.3) f(pn,T) is bounded and

thus it suffices to prove that f{pn,T} is closed. This assertien easily

fellows from the fact that the limit of a convergent sequence of T-periodic

functions is again T-periodic. -:inally, we have to show that 0 is an interior point of f{pn,T). Since

f(pn,T) is a subset of a finite dimensional space it is sufficient to prove

that for all .!!. E JRn wi th .!!. Y. .Q.

max { n T x I x ( r (p I T) } > 0 -- - n

Thio amounts to maximizing (cf. Lemma 6.3.1)

(6.3. 7)

over F(pn,T}, where Gis given by

(6.3.8) n-1

G(tl :=-T1 L n.Q<jl(T-t)

j=O J n

In view of Lemma 6.2.1 and the fact that llpn(D}fll :$ 1 if f E: F(pn,T), the

maximum in (6.3.7) equals

(6.3.9) T n-1

max {I L G(T)u(T)dT I u E Upn, llull :$ 1} . 0 j=O

Let the maximum in (6.3.9) be denoted by M.!l. If M.!l = 0 then for all k i Kpn

T n-1 (6.3.10) J L G(T)e-2k11'iT/T dT = 0 ,

0 j=O

141

since kt Kpn implies that the functions t,.....,.cos(2k'rrt/T) and t..._,sin(2k'1Tt/T)

belong to Upn.

Using the Fourier series of Qn and {6.3.9) we obtain for all k i Kpn

(6.3.11) n-1 n . < 2klf i/T) j

j~O p~ (2k'1Ti/T} 0 •

However, (6.3.11) implies that n0 = n1 = ... = nn-l = 0, which violates the

assumption ~ ~ Q. Consequently, M > 0 and hence 0 is an interior point of ~

r (pn,T}. D

We reeall {cf. Remark 2, p. 130) that at the end of Chapter 5 it was observed

that arm(o3

) contains straight line segments and therefore the set rmco3) is

not strictly convex. This notion will again be encountered presently; a

formal definition reads as fellows.

DEFINITION 6. 3.4. A eet A c lRn is aalled striatly aonvex if for all ~ E A

and all z E A the point ! (~+x_) is an interior point of A.

-Now it will be shown that r(pn,T) is strictly convex. To this endweneed

the following

THEOREM 6.3.5, Aseoaiated with each a= (a0 ,a 1, ••• ,a 1>T E êr(p ,T) there - n- n

existe a unique f E F(p ,T) euah that f(j) (0) =a. (j = O,l, ... ,n-1), n J

Moreover. thie unique f is a perfeat !-epline correeponding to the operator

pn(D).

PROOF. We note that the existence of f is an immediate consequence of the

compactnessof r{pn 1 T). So the uniqueness remains to be proved. Since

a E 3f(pn,T) there exists an ~~ Q such that

So, if f e F{pn,T) is a function with f(j) (0) aj (j 0 ,1, ••• , n-1) then

in view of (6.3. 7) and (6.3,9)

T T

(6.3.12) T f G('!)p (D)f('!)d'! =max{J G('!)U(T)d'! I u e Upn' llull s 1} . ~~ n

0 0

142

We reeall that this problem bas already been dealt with in Chapter 4 (cf.

pp. 97, 98)~ there it is shown that the maximum in (6.3.12) equals

T I IG(T) -g(-rlld-r

0

for an appropriate function g E Ker(pn,T) •

Hence, if the maximum value is attained at u0 E Upn with llu011 s 1, then

T

I IG(T) -g(-rlld-r

0

T

f (G(T) - g(T)) u0

(t) d't

0

T

s I IG(T) - g(T) ld-r

0

Therefore u0 (t} = sgn(G(t) -g(t)} (a.e.) on the set {O < t <TI G(t) J'g(t)}.

Note that the Fourier coefficients in the Fourier series of G(t) - g(t) are

not all zero. Consequently, the function G- g is a nontrivial function on

(O,T) and, because it is analytic on (O,T), it has finitely many zeros in

this intervaL It follows that u0 is essentially unique and jumps from ± 1

to +1 at the :zeros of G-g. Since pn(D)f(t) = u0 (t) (a.e.) on (O,T) and

f E F(p ,T), the function fis uniquely determined. Finally, the assertien n

that f is a perfect t-spline corresponding to the operator pn(D) is an

immediate consequence of Definition 1.3.5 and the properties of u0 •

Theerem 6.3.5 may now be used to prove

THEOREM 6.3.6. f(pn,T} is a etriatZy aonve~ eet.

PROOF. !t follows from the proof of Theorem 6.3.5 that for any n J' 0 the T "' - -

maximum of the linear function ,!o-+.!J..! on r(pn,T) is attained at a unique

point a E êf(p ,T). Now let a E 3f(p ,T),! E ar(p ,T) and assume that also - ...,n - n n T

~ (~ + ..ê) E ar(pn 1T). There exists a linear function ,!....,..n0 ,! attaining its

maximum on r (pm,T) at i(~+ ..ê) • Since .;:; ( j (~ + ..ê)) = ! c.;:; ~ + ~ !l I this

functional also attains its maximum at ~ and !· Because of the uniqueness

one bas~= l• and in view of Definition 6.3.4 it fellows that f(pn,T} is

strictly convex.

From the proof of Lemma 6.2.2 we may conclude ~at for every f E F(pn,T) a

function g E Ker(pn,T) exists such that f- g E F(pn,T). Consequently, if

pk(D) (0 s k s n-1) has the property that Ker(pn 1 T) c Ker(pk), then

0

0

maximizing I!Pk (D) fll on F(pn 1T) is equivalent to maximizing IIPk (D) fll on

F(p 1T). It turns out that T-periodic perfect t -splines are extremal with n

respect to this problem. Detailed information is given in the following

143

THEOREM 6.3.7. Let p En be a monie polynomiat and let pk(D) (0 $ k $ n-1) n n be a tinear differentiat operator of order k such that pk(D)g(t) = 0 (tE mJ

for aU g E Ker(pniT). Then there exists a T-periodic perfect t-spline s

corresponding to the operator pn(D) such that

(6.3.13) max llpk(D)fiJ = llpk(D)sll • fEF(pn 1T)

Moreover, if f 1 E F(pniT) also yietdS the maximum in (6.3.13} then there

exist numbers sE [OIT), cr E {-1 11} and a jUnction g 1 E Ker(pniT} such that

crs(t- ~) + g1

(t} (tEm> •

PROOF. If f E F(pniT) and if g E Ker(pn 1T) is the function satisfying

T T

J f (T) U(T) dT J g(T) U(T) dT (u E Upn} I

0 0

then f- g E F(pn,T). By Lemma 6.3.1 one has

f(t} g(t} + ~ r p (t-T)p (D)f(T)dT n n

(tEm) •

0

In view of this and taking into account the periodicity of f1 it follows

that

T

pk (D) f(O} = ~ f pk (D) pn (T- T)pn (D) f (T) dT

0

and therefore

(6.3.14} max IPk(D}f(O} I = _max IPk(D}f(O) I • fEF(pn 1T) fEF(pniT)

we abserve that maxiruizing IPk(D)f(O} I on F(pn,T) is equivalent to ~axi­mizing a linear function on f(p 1T) 1 the maximum of which is attained at a

n

144

boundary point of r(pn,T). By Theerem 6.3.5 there is a unique s ~ F(pn 1 T),

s being a T-periodic perfect t-spline eerrasponding to the operator pn{D),

such that

Hence, by (6.3.14),

pk(D)s(O)

_max Jpk(D)f(Oll f~F(pn,T)

_ max llpk (D) fll feF(pn,T)

This establishes the first part of the theorem.

max llpk {D) fll feF(pn 1T)

Now let f 1 e F(pn,T) yield the maximum in {6.3.13), let the numbers

~ e [O,T), cr e {-1,1} be such that

and let g1 e Ker{pn,T) be such that

Then by the equality crpk(Dlf 1 (~) = pk{D)s(O) and the uniqueness of s one

has s (t) == crf1 (t + 0 + g1 (t) (t E JR), which is equivalent to the assertien

in the secend part of the theorem.

k - n The problem of maximizing liD fll (k = O,l, ••• ,n-1) on F(D ,Tl has been in-

vestigated by Northcott [46]. He proved .that perfect Euler splines are ex­

tremal with respect to this problem. A generalization of Northeett's result.

to the nonpolynomial caseforsome specific operators pn(D) is due to

Golomb [19]. In this section we shall prove that perfect Euler l-splines

0

are extremal for specific functionals of the form f......,. 11 (apk (D) +bpk+l {D)) f!l ,

provided the polynomial pn has a few particular properties to he specified

in the following theorem.

THEOREM 6.4.1. Let p ~ 'Ir (n;:: 2) be a monia polynomiat having only reat n n

zeros and let pn{Ol = 0, FurthePmare~ let pk+l ~ 'lfk+l (k s n-2) be any monia

polynomiat that is a divisor of pn~ and let pk ~ 1rk be suah that

145

pk+1 {t)

one has

(t-a)pk(t) (tE lR). Then for any (a,b/ E lR2 with (a,b) ,;, (0,0)

(6.4.1) _max 11 (apk (D) +bpk+l (D)) fll fEF(pn,T)

where f(p ;T/2,•) denotes the perfeat Eulerf.-spline as n

in Defini-

tion 3.2.12. Moreover, if a function E F(p ,T) yields the maximum in n

( 6. 4 .1) , then

(6.4.2) (t E JR)

for appropriate aonstants ~ E [O,T), and a E {-1,1}.

PROOF. According to the definition of Kpn (cf. p. 137) and the assumption

on the zerosof pn' we conclude that Kpn = {O} and hence, by (6.3.3), Pn

satisfies the differential equation

(6.4.3) p (DJP (t) =-1 n n (0 < t < T) •

Let the tunetion F be defined by

(6.4.4) (t E lR) •

In view of (6.4.3) and the assumption on pk and pk+l

(6.4.5) p k(D) F(t) = (ai +b(D- ai))p (D) P (t) = -a +ab , n- n n

where pn-k(t) := pn(t)/pk(t) (tE lR).

Furthermore, since F E w<n-k- 2) (lR,T) one has

(6.4.6) F(j) (0) = F(j) (T) (j O,l, ••• ,n-k-3)

It fellows from Lemma 6.3.1 that for all f E F(pn,T) and t E lR

(6.4. 7) (apk(D) +bpk+l (D))f(t) 1

=-T I F(t- T)pn (D) f(T) dT .

0

In view of this and taking into account Lemma 6.2.2, we conclude that the

problem of maximizing ll(apk(D) +bpk+l (D))fll on F(pn,T) is equivalent to the

problem of maximizing

(6.4.8) 4- I F(T-T}U(T}dT

0

146

wi th respect to those functions u E Up for which llu 11 s: 1 • n

Since pn has only real zeros with pn(O) = 0 the set Ker(pn,T) only contains

constant functions and therefore (cf. (6.2.8})

T

Upn ={u € L(X)(lR,T) I J u('r)dT o} . 0

Consequently,

(6.4.9)

sup H J F(T-T)U(T)dT I u E L .. ([O,T]), I u(T}dT .. o}. lluii[O,T]S:l O O

If there exists a constant function d such that

(6.4.10) r sgn(F{T- T) - d) dT

0

r sgn(F(T) - d)dT

0

0 ,

and d coincides with F at at most finitely many points, then by Theerem -1 T -

6.3.5 the supremum in (6.4.3) equals T 0J !F(T)- dldT.

In view of the remark on p. 98 such a constant function d exists if one

shows that for any constant function d, F- d has fini tely many ze ros. This

can be provedas fellows.

Writing Hd(t) := F(t) -d and using (6.4.5) one has

(0 < t < T)

with (cf. (6.4.6))

H (j) (0) = H (j) (T) d d

(j = 0,1, ••. ,n-k-3) •

Because of (6.4.11) Hd coincides on (O,T) with an analytic function. If Hd

would have infinitely many zeros on (O,T) then Hd would vanish identically

on (O,T}. Hence F(t) = d (0 < t < T) and therefore, for all f E F(pn,T), we

would have (cf. (6.4. 7})

(apk(D) +bpk+l(D))f(t} = ~ r pn(D)f(T)dT

0

0 ,

since pn (0) 0. This, of course, is not true. Hence Bd has finitely many

147

zeros in (O,T) and so a constant d can be found satisfying (6.4.10).

Our next purpose is to estimate the number of zeros of Hd in (O,T) by using

the Budan-Fourier theerem 1.4.14.

Now letpn-k(D) (D-Àn-ki)(D-Àn-k-1 !) (D-À 1 I), Di (D Àii)

(i= l, .•. ,n-k} and o[i] = DiDi_1 ••• o1

(i= l, ... ,n-k). One easily verifies

that

(6.4.12) (T) (i 1,2, ••• ,n-k-3) •

If (cf. (6.4.11)} pn-k(D)Hd(t) ~ 0 then in view of the Budan-Fourier

theerem 1.4.14 and (6.4.12)

+ S (Hd (T)

way it can be shown that

On theether hand, if pn-k(D)Hd(t) = 0 and D[n-k-l]Hd(O+)D[n-k-l]Hd(T-)

then D[n-k-l]Ha(t) = 0 (0 < t < T). We assert that in this case

0,

D[n-k-2]H (0+)D[n-k-2]H (T-) # 0. If not, then D[n-k- 2]Hd(t) = 0 (0 < t < T) d d

and so in view of (6.4.12) the function Hd would coincide on (O,T} with a

function in Ker(pn,T}. Consequently, Hd(t) = y (0 < t < T} for some constant

y. However, we have shown already that this cannot be the case. Hence

D[n-k- 2]a {O+}D[n-k- 2]a (T-) I 0. Using Theerem 1.4.14 again we then have d d

from (6.4.12)

n-k-2 Z( (Di}

1 ,Hd, (O,T}) = 0 .

We conclude that one of the following four possibilities occurs: Hd has no

zerosin (O,T), Hd has exactly one zero in (O,T), Hd has exactly two dis­

tinet simple zeros in {O,T), and Hd has exactly one zero of multiplicity two

in (0,T). However, since by (6.4.10) 0

JT sgn(Hd(T})dT 0 the function Hd

has exactly one simple zero in (O,T) located at t = T/2 or two simple zeros

in (O,T) a distance T/2 apart. Therefore, if u denotes the periadie exten­

sion of a function u that maximizes (6.4.9), then Ü(t tT/2) = - Ü(t). Con­

sequently, the "p (D)-derivative" of a function f 1 E F(p ,T) satisfying n n

148

Pn(D)f1 (t) = Ü(t) (a,e,) jumps from ± 1 to:;: 1 at equally spaeed points a

distance T/2 apart. In view of the definition of a perfect Euler ~-spline

(cf. Definition 3.2.12), we conclude that f 1 (t) = aE(pn;T/2,t-t) (t e JR)

forsome constants a e {-1,1} and t E[O,T). This proves the first part of

the theorem.

The second part of the theerem fellows fiom the observation that the pericd­

ie continuatien of every function u that maximizes (6.4.7) jumps from ± 1 to

:;: 1 at equally spaeed knots a distance T/2 apart. So each function

f 0 e F(pn,T) which yields the maximum in (6,4.1) and for which pn(D)f0 (t)

= u(t) (0 < t <Tl has the form (6.4.2). 0

REMARK 1. In case pn has only real zeros and pn(O) # O, then (6.4.7) has to

be maximized without the restrietion 0JT u(T)dT = 0. Hence u(t) = sgn(F(t)}.

It is easy to construct an example such that F(t) > 0 (te (O,T)). The eer­

responding extrema! function then satisfies pn(D)f(t) = 1 and so f(t) = -1 = pn (0) (t E JR); this function is not a perfect Euler t-spline.

REMARK 2. If not all zeros of p are real then the problem as solved in n .

Theerem 6.4.1 bacomes more complicated. This is due to the fact that in

this case an interval of disconjugacy for pn(D) is finite, and the length

of the interval [O,T] is then of importance. If T is small enough one may

expect that results similar to Theerem 6.4.1 hold.

As an illustration of the preceding results a parametrization of the set ~ 3 ~ 3

3r(D ,1) will be given. In view of Lemma 6.3.1 a function f € F(D ,1) can

be written as

(6.5.1)

1

f(t) = J P3(t-T)f 111 (T)dT,

0

where (cf. (6.3.2))

2k'lfit p3 (t) = L .;;;.e_--=­

k~ (2k1Ti) 3 I

{t € JR) •

According to Remark 3 ( cf. p. 139 l , apart from a mul tiplication factor - 6,

149

P3 coincides on (0,1) with the Bernoulli polynomial of degree three, i.e.,

- ...!.._ t 12

(0 :::; t :::; 1) •

Hence, if n0 , n1

, and n2 are real numbers with <n0

,n1,n 2) f (0,0,0) then

(cf. (6.3.7) and (6.3.8))

(6.5.2)

where

t F(1

0

T)f 111 (T)dT,

F(t) (0 :::; t :::; 1) •

1 Consequently FE n 3 and 0f F(T)dt = 0. In order to maximize (6.5.2) on

3 r(D ,1) one has to compute (cf. (6.4.9))

1

max { f F(1-T)u(t)dt I u E

0

1

(lR,T), J U (T) dt

0

0, 11 uil :::; 1} .

1 -that this maximum equals 0f IF(t) -dld"C, where dis a constant

1 - -We reeall

such that 0f sgn (F ("C) - d) dt = 0. Since F- d E rr 3 i t has at most three sign

changes in (0,1) 1 evidently it bas at least one sign change in (0,1). 0 Defining t + := ~ ( 1 + sgn (t)) , we now may wri te

with 0:::; ~;: 1 s ~;: 2 s 1;3

s 1, ~;: 1 -1; 2 + !; the conditions on 1; 1 , 1; 2 , and ~;: 3 fellow from the fact that / 1 sgn(F("C) -d))dt = 0. The functions

~ 3 f0

E F(D ,1) for which fQ"(t) = ± sgn(F(l-t) -d) (0 < t < 1) are then

given by (cf. (6.5.1))

1

± f P3 (t-"C)(1-2(1

0

By a straightforward computation we get for i = 0,1,2

(6.5.3)

where (cf. Remark 3, p. 139) P2

; P;, P3 = P~, and P4 is given by

150

(6.5.4) P == - 1.. t4 1 3 1 2 1 4 (t) 24 + 12 t - 24 t + 720 (0 $ t $ 1) •

Since f0

maximizes (6~5,2) the point (f0

(0),f0(0),fÖ(O))T € ar(o3,1). On

the other hand, for any ~ 1 , ~ 2 , ~ 3 with 0 s ~ 1 s ~ 2 S ~ 3 S 1 and

~~-~ 2 +~ 3 =! the expressionsin the right-hand side of (6,5.3,) yield a - 3 ; point of ar(D ,1), which assertien may be shown by choosing appropriate

values for n1 , n2

and n3

in (6.5.2);we omit details. Therefore tihe right-hand - 3 side of (6.5.3) determines a parametrization of ar(o ,1).

Taking ~ 3 = 1 and ~ 2 = ~ 1 + l and using (6.5.3) we obtain boundary points

of the form

where 0 S ~~ s ~.

Now, by (6.5.4) and (3.2.20)

1 1 3 1 2 1 · E 3 1 2(P4

(t) -P4(t+ 2)) = 6 t -8 t +

192 =- (D 12,t) (0 s t s i> •

3 I 3 1 :ro" 3 1 T Consequently, for all t € :R the points (t(D d,t) ,E (D n,t) ,c (D ;2,t)) - 3 lie on the boundary of f(D ,1).

151

7. PERFECT t-SPLINES ANV THE LANVAU PROBLEM ON THE HALF LINE

7. 7. Intnoduction and ~umm~y

Landau problems are described in the introductory sectien of Chapter 5.

Given a closed subinterval J of ~ and given a linear differential operator

pn(D) of order n, the problem is todetermine the best possible upper bound

for llpk(D}fiiJ with respect to a specific subset Fm(pn,J) of W(n) (J) (cf.

(5.1.2)), where pk(D) is a given linear differentlal operator of order

k ~ n-1. Functions for which the best possible upper bound is attained are

called extremal functions (cf. p. 103). For second order differentlal

operators p 2 (D) it is shown there that perfect Euler t-splines are extremal

in the full-line case and one-sided perfect Euler t-splines are extremal in

the half-line case.

In this chapter we shall deal exclusively with the half~line case, and a

Landau problem then is an extremal problem with respect to the space

W(n) (~~) endewed with the sup norm. Of course, one of the main objectives

in solving the half-line case Landau problem is to determine extremal

functions1 in view of this, we shall investigate how perfect t-splines are

involved.

Throughout this chapter, the menie polynomial pn E ~n associated with the

differential operator pn(D) is assumed to have only real zeros; this is

done to avoid complications with respect to disconjugacy of pn(D).

We reeall that Sectien 5.4 deals with the relation between the Landau

problem on the half line and the set r+(p ) as defined on p. 110. It is m n shown there that maximizing 118

0f+8 1f' + ••• +8n_

1f(n-l)ll+ on !pn,IR~) is

equivalent to maximizing a linear function on r+cp ) • Therefore the problem rn n

of determining extremal functionsisintimately connected with the problem

of deterrnining functions f E F (p ,IR~) such that for some ~ E ~~ the point ' (n-1) T m + n

(f(~) ,f (~) , ••• ,f (~)) E ilrm(pn), the boundary of (pn). This gives

rise to the following interesting extremal problem: given

.<!_ == (a0 ,a1 , ••• ,an-l)T E ~n, determine f E Frn(pn,IR;) wHh f(i) (0) a1

(i== O,l, ••• ,n-1) and such that

152

(7.1.1)

Determining a function Î satisfying (7.1.1) is one of the main purposes of

this chapter.

A brief review of the contents of the various sections now fol:lows. In

Sectien 7.2 some preliminary material is collected. Starting ~~om the set

Pr {p ) of perfect .C-splines ha ving all knots in ~, we. introduce the sets P (p ), n r,a n p(e:} {p) and P(e:) (p). Propertiesof P {p) 1 p(e:) (p) 1 P (p) and P(e:) (p) 1

r n r,a n r n r n r 1 a n r,a n due to Karlin [2J] and Cavaretta [10], are given. Apart from this, we-derive

some compactness properties of P (p) and P(e:) (p }. So-called representa-r n r n tien theorems are dealt with in Sectien 7.3. A few fundamental results of

this type, due to Karlin and Studden [25] and Karlin [23], are given for a

set of functions spanned by a Chebyshev system, and for the set Pr(pn).

Our objective is to derive a representation theerem for the set Pr,a(pn);

this is done via a representation theerem for P (~) (p ) , the set of "e:-r,a n approximate" perfect .C -splines. Sectien 7. 4 is de~ted to the problem of

obtaining a function f satisfying {7.1.1), Wedetermine f as the limit of

a sequence of perfect .C-splines1 each perfect t-spline of the sequence is + a salution of an extremal problem similar to (7.1.1), ~O then being

replaced by an appropriate finite interval. These extremal .C-splines con­

verging to f exhibit a specific oscillation behaviour. In order to prove

their existence weneed the representation theerem of Sectien 7.3.

Finally, Sectien 7.5 contains, among ether 3

things, two examples. The first

one concerns the operator D , and a complete description 4 is given. The secend example deals with the operator D

+ 3 of the set arm(D )

Even for this

simple operator, it is difficult to determine an extremal function by

analytica! means. Instead, a sequence of perfect t-splines converging to

an extremal function is obtained numerically.

153

We reeall (cf. Definition 1.3.5) that perfect!-splines are introduced in

Chapter 1. In conneetion with the Landau problem on the half line only

perfect t -splines wi th knots in JR~ are of interest. In this chapter a

function g E Ker(pnl will also be called a perfect !-spline corresponding

to the operator pn(D); gis said to be a perfect L-spline having -1 knots.

Fora given integer r ~ -1 the set Pr(pnl of perfect !-splines is now

defined as fellows.

DEFINITION 7.2.1. Let pn E ~n be a manie polynomial having only real zeros

and Zet r+1 E N0 • Then P r (pnl is the set of perfect t-spZines cor>responding

to the operator> pn(D) and having at most r knots, all Zocated in JR~.

It fellows from the preceding definition that

If s Pr(pn) has precisely k knots x 11 x 21 ••• 1 ~ with k E ~01 then scan

be represented in the form

(7 .2.1) s (t)

t k

g{t) + c f (~{T) + 2 I i=l

{ -1) i (j) {T - X,)) dT + l. '

( t E JR) 1

0

where gE Ker(pn) 1 c E JR with c # 0 1 and where (j) is the fundamental func­

tion corresponding to pn(D).

In what fellows, perfect t-splines in Pr(pn) will be needed for which the

derivatives at 0 1 up to the order n-1 1 are prescribed. This leads to

DEFINITION 7.2.2. Given ~= (cr.01 cr. 11 ,,,,cr.n-l)T E JRn, the set P (p) is r 1E:_ n

defined by

P (p ) := {s E Pr(pn) I s (i) (0) r,~ n

Q 1 1 1 • • • 1 n-1}

The purpose of this sectien is to derive some properties of Pr(pn) and

P (p ) for later use. r 1 cr. n

we ëmphasize that throughout this chapter p E n is a monic polynomial n n

with real zeros only.

We cbserve that Pr(pn) is nota linear space of functions; it depends non­

linearly on n+r+l parameters. It is known (cf. Karlin [22]) that Pr(pn)

154

is a solvent family of ordernon ~; (cf. Definition 1.4.20), i.e., if

f € AC(n) (~;) and if 0 !> ~ 0 !> ~ 1 s ••• s ~n+r with ~i < ~i+n (i •O,t, ••• ,r)

then a function s € Pr(pn) exists such that (cf. p. 4 for notation}

(7.2.2)

An important property of perfect l-splines is contained in the following

THEOREM 7.2.3 (Karlin [22]). Let a> 0~ let f E w(n) ([O,a])~ and Zet

n+r+1 points 0 s ~Os ~ 1 s ... s ~n+r sa be given suah tha~ ~i< ~i+n (i = 0,1, ••• ,rJ. Then a perfeat l-spline s E P (p ) e~sts satisf.Ying r n (7.2.2) and having its knots in (O,a), Mor>eover~ any perfeat l-spUne

s E Pr(pn) satisf.Ying (7.2.2) has the prope'l'ty

(7.2.3) lp (DlsH s Hp (D)f"[o ] • n n ,a

The interpolating perfect l-spline of Theerem 7.2.3 is in general not

uniquely determined. However, the following result holds.

LEMMA 7.2.4 (Cavaretta [10]). Let ö s ~; 0 S ~l s ••• si; 1

be n+r n+r-points ~ith ~i< ~i+n (i= O,l, ..• ,r-1). Phen a nontPiviaZ perfeat l-spline

s E Pr(pn) e~sts~ uniquely dete'Pmined up toa multipliaative oonstant,

having preaisely r knots and~ith the pr>operty ~<~;0 ,~ 1 , ••• ,~n+r- 1 l O.

Moreover, the knots x 1 < x 2 < ••• < xr of any suah s satisfY the inequali­

ties

(7.2.4) (i O,l, ••• ,r-1).

In fact, in proving Lemma 7.2.4 Cavaretta dealt only with the polynomial

case p (D) = Dn. As the proef of Lemma 7.2.4 is quite similar to n

Cavaretta's, it will be omitted here.

The main difficulty in dealing with perfect l-splines arises from the fact

that Pr(pn) is nota unisolvent family (cf. Definition 1.4.20), i.e., a

function s € Pr(pn) satisfying (7.2.2) is in general not uniquely deter­

mined. In order to evereome this difficulty, as a standard tool one uses

the so-called "e-approximate" perfect l-splines. These functions will be

defined by means of a specific positive linear operator.

(n) + If f E AC (JR

0) then by Taylor's formula (cf. Lemma 1.4.4) f can be

written in the farm

f(t) g(t) + J (p+{t--r)pn(D)f(T)dr

0

(t ~ 0) ,

where gE Ker(pn} satisfies g(i) (0)

Let f be defined by

f(i) (0) (i 0 I 11 • • • 1 n-1} •

f(t) '"' ( f(t) g(t}

(t ~ 0) 1

(t < 0)

Now for any E: > 0 the "smoothing" operator

(7 .2.5) (T f) (t) := E:

is defined by

(t E JR)

155

for functions f E AC{n) (JR~) for which the integral in (7.2.5) converges.

Note that if pn(D)f(t) = 0 (t > 0}, then pn(D)f(t) = 0 for all tE JR1 and

therefore TEf E Ker(pn).

We are now ready to define the sets

perfect t -splines.

(E) (pn) and P (E) (p ) of "E-approxirnate" r1!:!_ n

DEFINITION 7.2.5. Let E > O, let P (p) and P (p l be the sets of perfect r n r,~ n t-splines of Definitions 7.2.1 and 7.2.2. Furthërmore, Zet the operator TE

be given by (7.2.5). Then

P (E) (p ) := {T s r n E

S E

{T s I sE p (p )} E r,!::_ n

Just like P (p ) the set P (E) (p ) depends nonlinearly on n + r + 1 parameters. r n r n

The important difference between the sets P {p ) and P(E) (p ) is that r n r n P;E) (pn) is unisolvent (cf. p. 22) 1 whereas Pr(pn) is only a solvent farnily.

THEOREM 7.2.6 (Karlin [22]). The set + order n + r + 1 on JR0

• Moreover,

(E) ( ) • • l t Pr pn ~sa un~so ven farrrily of E p(E) (p ) satisfies

r n

156

~E(~O'~l'''''~n+rl = (yO,y1, ••• ,yn+rl 7~ whePe 0 s ~0 s ~1 s ··· s ~n+r and y

1 é m (i= O,l, ••• ,n+r) are given~ then se: depende aontinuously on the

parameters ~ 0 .~ 1 , ••• ,~n+r'Yo•Y 1 , ••• ,yn+r'

In Karlin's proof of Theerem 7.2.6, which is rather complicated, it is

assumed that pn (D) = Dn. However, as in t.he case of Lemma 7.2 .4 the proof

can also be carried through for a general differential opet:ator pn (D}, if

pn has only real zeros. We cbserve that a simpler proof of Theerem 7.2.6

can be given by making use of ideas of De Boor [4].

For later use we now derive some compactness properties of the sets

P;E) (pn) and Pr(pn) •

co • (€) LEMMA 7.2.7. Given E > 0~ let (s ) 1 be a sequenae ~n P (p }. If &,m m= r n there is an interval [a,b] c m; with a < b~ and a positive M suoh that

Is (t) I s M for all m é ~ and tE [a,b], then a subsequenoe (s lk"'-1 e:,m ( } e:,mk -exists converging to an s EPe: (p l suah that for i= 0,1,2, •••

E r n

lim s (i) (t) = s (i) (t) , k+oo E,mx E

uniformly in t on eVe17J bounded interval of :R.

PROOF. Sinces E P{e:) (p} we may write s T~sm with sm E Pr(pn). e:,m r n e:,m ~

Hence, according to (7.2.1),

(7.2.6)

t

( 7. 2. 7) h (t) m I (ql(T)

r . ) + 2 L (-1) J ql (T- x j) dT

j=l + m, , {t :?! 0) •

0

Here xm,l < xm, 2 < ••• < xm,r are the knots of sm; if sm has q knots with

q < r, then we take x +l = x +2 = . • . x = <» and cp+ ( t - x . l = 0 m,q m,q m,r m,J (tEm, j = q+l,q+2, ... ,rl.

Without loss of generality it may be assumed that lim~ x . ""- m,J

(j = 1,2, ••• ,r) with xj E lR~ or xj = ""• since there always is

of (hml having this property. We now define the function h by

=: xj

a subsequence

t

(7 .2.8) h(t) := J (~(T) 0

+ 2 I ( -1) j ~ ( T - x.)) dT + J

j=1 (t E JR) •

Since ~+(t-xm 1 j) -+ ~+(t-xj) (m-+ co) 1 uniformly int on every bounded

interval of lR 1 we also have

(7. 2.9) h (t) _,_ h(t) m

(m -+ co) 1

uniformly on every bounded interval of JR.

We preeeed by proving that the sequence (cm) is bounded. To this end we

take n+1 points ~O < ~ 1 < < ~n in [a 1 b] and consider the divided

difference s [p ;~ 0 ~~ 11 ••• 1 ~] (cf. Definition 1.4.25). In view of E 1 m n n

(7.2.6) and taking into account that TEgm E Ker(pn) it fellows that

(7. 2 .10) s [p '~o~~1~···~~ J =cT h [p '~o~~1~···~~ J. E 1 m n n m E m n n

Because of (7.2.9) and the definition of T (cf. (7.2.5)) one also has E

157

TEhm(t) -+ TEh(t) (m-+ co) 1 uniformly int on every bounded interval. Hence

(m _,_ co) •

Since (s [p ;~0 ~~ 11 ••• 1 ~ ])co 1

is a bounded sequence the boundedness of E 1 m n n m=

(cm) will fellow from (7.2.10) if one shows that the points ~ 0 ~~ 11 ••• ~~n

can betaken such that T h[p ;~ 0 ~~ 11 ••• 1 ~] I 0. By (7.2.8) and (7.2.5) one E n n

has for all t E JR

(7. 2 .11) p (D)T h(t) n E J e-(t-T)

2/E(1 + 2 I (-1)j(T-x.)~)dT 1

liiE j=1 J 0

where (t-x.)0

:= !(1 +sgn(t-x.)). J + J

So pn(D)TEh is an analytic function and 1 moreover 1 it is nontrivial 1 since

one easily verifies that

lim pn(D)TEh(t) I 0 . t-+oo

We conclude that pn(D)TEh has finitely many zerosineach bounded interval.

Consequently 1 points ~o~~ 1 ~···~~n exist such that TEh[pn•:o~~ 1 ~···~~n] I 0.

This proves the boundedness of (cm). Therefore 1 a convergent subsequence

of (cm) exists and we may as well asssume that cm -+ c E JR (m -+ co) • It

fellows from the identity TEsm = TEgm + cm TE hm (m E iJl and the fact that

158

(T s ) and (c T h ) are bounded sequences on [a,b] that (T~ gm) is e: m m e: m (n- 1) ~

bounded on [a,b]. Since Tg € Ker(p} and ~~~·, ••• ,~ forma basis e: m n for Ker(pn} we may write

( t € :R) •

By taking n distinct points ~ 1 < ~ 2 < ••• < ~n in [a,b] we have

1:, Since ~~~· , ••• ,~<n- 1 ) is a Chebyshev system on [a,b] (cf. Example 1, p. 19).

Consequently, the coefficients a " (t: O,l, ••• ,n-1) in (7.2.12) are m,"'

uniqualy determined by the linear equations

(i l, •.• ,n) •

Because of the boundedness of (Te:gm} on [a,b] we deduce that the sequences

0,1, ••• ,n-1) are bounded. Hence, there is a subsequence of

(Te:gm) converging to a function g € Ker(pn), uniformly on every bounded

interval. We conclude that a subsequence (Te:~):=l exists such that

(7.2.13} lim T Sm. (t) k-+<» e: K

uniformly int on every bounded interval; in (7.2.13) at :: li~-+<» affik,t

and h is given by (7.2.8).

Since for every i € :N one has Di Te h~ (t} ->-Di Te h(t) (m->- =), uniformly in

t on every bounded interval of :R, the secend part of the theerem easily

fellows from (7.2.13). 0

In a similar way one may prove the following

LEMMA 7.2.8. Let (s >""1 be a aequenae in P (p ), If thePe ia an interval m r n [a,b] eR~ with a< b, and a poaitive M auah that lsm(t) Is M for all

mEN and t E [a,b], then a subsequenae (smkl:=l exists converging to a

funetion s E P (p l suah that for i 0, 1, •.. , n-1 r n

lim s (i) (t) = s (i) (t) 1 k-><» mk

uniformly in t on every bounded intewal of lR.

The next result is intimately connected with Lemma 7.2.7.

THEOREM 7.2.9. Let (~m):=l be a sequence of positive numbers with

limm+oo ~m = o, and let (s 8 m):=l be the assoaiated sequenee funations

s 8m E P;Eml (pn). If there is an interval [a,b] c]R~ and a positive M

suah that ls8 (t) I ~ M aZl mEN and tE [a,b], then a subsequenae m

159

(s~mkl== 1 exists converging to an sE Pr(pn) suah that for i 0111•••1n-1

S (i) (t) 1

uniformZy in t on every bounded intewal of lR.

PROOF. The proof follows the same pattern as that of Lemma 7.2.7; the only

difference is that we have to use the observation that for j 1, 2 1 ••• ,r

Cjl (t- x.) , + J

uniformly in t on every bounded interval of lR. 0

We reeall that P(E) (p ) is not a linear space 1 so if s1

E (E) (p } and (E) r n (E) n

s2

EP (p) then, in general, i<s 1 +s2

l i P (p ). However, for specific r n · r n pairs s

1, s

2 the existence of an s E p~E} (pn) may be shown such that

i (s 1 + s2

l shares some properties with s. Details are given in the next

lemma.

LEMMA 7.2.10. Let s 1 E

~1 (~O'~l'''''~n+r-1) 0 ~ ~0 ~ ~1 ~ ••• ~

there exists a unique s

(E) (p l and s2

E P(E) (p l satisfy the condition n r n

<~0 .~ 1 •••. ,~n+r- 1 l forsome sequence of points

• Then for every t 0 E lR~ with (t0 l 'I s 2 (t0l

p(E) (p) having the properties, r n

160

{

s ct0 l = ~ cs 1 Ct0 > + s 2 Ct0})

(7 .2.14)

~(~O'~l'···•~n+r-1) ~1 (~0'~1'''''~n+r-1l

(7.2.15l min {s1

(t) ,s2

(t)} < s(t) < max {s1 (tl, (t)}

+' 1 l + . h J' JO!' ac" t JR0

W'l-t s 1 (t) r s 2 (tl .

PROOF. In view of Theerem 7.2.6 there exists a unique sE P(e:) (p l sati5-r n

fyl.ng t;O,~l' · • · '~n+r-1) = ::1 (~0'~1' · • • '~n+r-1) and s{tO) = = ~(5 1 Ct0 l +s 2 <t0)). Soit remain5 to prove (7.2.15) for all

ti {t0 ,~0 .~ 1 , ... ,~n+r-l}. Since 5 I 51 and s ;i s 2 , by Theorem, 7.2.6, s

coi.ncides with s 1 and s 2 only at the n+r points ~0 ,~ 1 , •.. ,~n+r- 1 • Further,

it is no restrietion to assume the situation as sketched in Figure 1.

Figure 1.

Here t 1 and t 2 are two consecutive elementsof {~0 ,~ 1 , ••• ,~n+r- 1 }, t 1 being a zero of s 1 - of odd multiplicity and t

2 being a zero of s 1 - of

even multiplicity. Since ~Ct;0 ,; 1 , ••• ,~n+r-l) = !1 (;O,~l''''';n+r-1)

= ;!! 2 Ct;0 ,t; 1 , ... ,;n+r-ll' t 1 is a zero of s-s1 , s-s2 , and s 1 -s2 of the

same multiplicity. This as5ertion, obviously, also hold5 for t 2 .

consequently, as 52

Ct0

l < 5(t0) < s1

(t0

l it follows that s 1 {t) < s(t) <

< s2

(t) (t1

< t < t2

) and s1

(t) < s (t) < s2

(t) (t > t2l. This ascertains

(7.2.15) and the lemma is proved. IJ

161

7. 3. A Jz.eptr..e-óen:t.cttlon .theOJz.ern 60!!. .the ;.,et P (p l r,~ n

It is well known (cf. Karlin and Studden [25, p. 293] that Chebyshev poly­

nomials are "extremal" with respect to the problem of maxirolzing llp'll[-l,lJ

on the set of polynomials p in rr n for which 11 pil [ _1 , 1] s 1 . The Chebyshev

polynomials are characterized by the fact that they oscillate a maximum

number of times between -IIPII[-1 , 1] en IIPII[- 1 , 1], i.e. points -1 T1 <

< T2 < ••• < Tn+1 = 1 exist such that IIPII[- 1, 1] = lp(Ti) I and p(Ti+1} = =- p (T 1l (i = 1, 2, ••• 1n) • There are many problems in approximation theory

the corresponding extremal functions of which are characterized by a

specific oscillation behaviour (cf. Karlin and Studden [25, Chapter IX]).

In many such cases characterizations of extremal functions can be obtained

by a suitable application of a so-called representation theorem. These

representation theorems have in common that they ensure the existence of

specific functions that oscillate a prescribed number of times between two

given continuous functions. As an illustration we give a representation

theerem for functions spanned by a Chebyshev system.

THEOREM 7.3.1 (Karlin and Studden [251 p. 72]). Let ~ 0 ,~ 1 1•••1Wn be a

Chebyshev system on [a1b] and let f E C([alb]), gE C([a,b]) be suah that

a funation v of the form v = l:~=O wi wi e:dsts satisfying g(t) < v(t) < f(t)

(t E [alb]). Then there e:dst unique fune~tions ~ = r:=o ~i ~i and ,..n - • h h • v "'i=O w i ~i w~ t t e propert-z-es

i) g(t) s ~(t) ::;; f (t) (t € [a,b]) I

iil there e:dst n + 1 points a ::;; lt < !2 < ••• < ln+1 s b such

(7. 3.1) that

{ •<r11 (i odd) ,

~(!,i) = f (T.) (i even)

-~ I

i) g(t) s ~(t) ::;; f(t) (t € [a,b]) I

ii) there e:dst n + 1 points a ::;; Tl < T2 < ••• < Tn+l $ b suah

(7. 3. 2) that

{·'~') (i even) , ~(:;,,

~ f{Ti) (i odd)

162

A representation theerem for functions spanned by a Chebyshev system and

satisfying some dppropriate boundary conditions is given by Pinkus [47].

Beside these results there is a representation theorem for pertect i­

splines; before we go into this, it is convenient to have the following

definition available.

DEFINITION 7. 3. 2. Let f and g be two j'unations dBfined on an interval J

with the property g(t) < f(t} (t E J), A j'unation h is said to osaiUate k

times betiJeen f and g on J if g(t) s h(t} s f(t} (t E J) and if there

exist at least k + 1 points T 1 < T 2 < ••• < tk+l in J such that

i} h (T o) . { .,,,,

l. f(T i}

(i oddl ,

(i even) ,

(7. 3. 3) or

ii) h (Ti} -f .,,,,

f(T o} l.

(i even) ,

(i oddl •

It follows that if h oscillates k times it also oscillates k- 1 times.

If i) of (7.3.3) applies then the set {t1,t2 , ••• ,tk+l} is called a set of

k + 1 osc:iUation points of hof type I; similarly, !f ii) of (7.3.3) holds

then {t 1 ,t2 , ... ,Tk+l} is called a set of k+l osaiUàtion points of hof

type II.

THEOREM 7.3.3 (Karlin [22]). Let a> 0 and let f E C([O,a]), g € C([O,a])

be suah that a function s ~ Pr(pn) exists satisfying g(t) < s(t) < f(t)

(t ~ [O,a]). Then funations s ~ P (p ) and; E P (p } exist osaillating - r n r n

n + r times between f and g on [O,a] and suah that ! has a set of n + r + 1

m~aiZZation points of type I and s has a set of n + r + 1 osaiUation points

of type II.

The main purpose of this sectien is to derive a representation theerem for

the set P (p ). This result will be used in Sectien 7,4 to show the r,a n existence of appropriate extremal functions. In order to achieve this we

first deduce a representation theorem for the set P ( e:) {p ) as introduced r,!!_ n

in Definition 7.2.5.

163

THEOREM 7 .3.4. Let Cl E lRn~ a > 0~ and let f E C([O,a]), g E C([O,a]) be

suah that an sE P(ZJ (p l e:x:ists satisfying g(t) < s(t) < f(t) (tE [O,a]J. . .r,~ n (e:) ~ (e:) • •

Then un1-que funat'l-ons s E P (p l and s E P (p l e:cz,st osa-z. r "..", r ,ex n r ,a n times between f and g on [O,a] suah that z (SJ-has a set of r + 1 osoiZlation

points of type I (II).

. 11 . ~ (€:) ) PROOF. F~rst we sha prove the ex~stence of~ and s. LetsE Pr,a(pn

satisfy g(t) < s(t) < f(t) (tE [O,a]). According to Theerem 7.2.6 for

every sequence of n+r+l points 0 = 1;:0 1;: 1 = ••• = s !;;n s Çn+l s

s ... s !;;n+r a and for every À E lR there exists a unique v E (c) (pn)

such that

i)

ii) (7 .3.4)

v(j-1) (!;; ) n+r

À ,

where j E N is determined by !;; + . < !;; + .+ 1 n r-J n r-J

= l;n+r a.

In view of the unisolvency of P~c~(p ) (cf. Theerem 7.2.6) it fellows that ('-1) . ,_ n

v = s if À s J (I; ) = s (J-ll (a), since then _!(!;:0

,1;1

, .. · ,i;n+r) = n+r ( . _ 1)

= ~(ç0 ,ç 1 , ••• ,Çn+r); the number v J (a) is denoted by Àa. Note that Àa

depends on j and therefore on the sequence ç0

,i;1

, ••• ,l;n+r' Wedefine the T r+l

vector~= (h0 ,h1 , ••• ,hr) E lR by

(7.3.5)

Hence,

by

a and h. ;::: 0 (i ~

V := {~ E lRr+1 I I hi i=O

(i O,l, ... ,r).

O,l, ••. ,r). Let the simplex V be defined

a, h1

;::: 0, i= O,l, .•. ,r} .

The function v satisfying (7.3.4} will be denoted by v(~,À,•).

In view of Theerem 7.2.6 it then fellows that (~ 1 À,t)~v(~,À,t) is con-

tinuous. Since by assumption g(t) < v(~,Àa,tl s(t) < f(t) (t E [O,a])

for every ~ E V, Àa is an interior point of the set

Ah : {À E lR I g(t) s v(~1 À 1 t) s f(t), t E [O,a]'}

It fellows from the compactnessof P;c) (pn) (cf. Lemma 7.2.7) that ~is bounded. We next define

164

(7. 3 .6) := sup fln .

* Our purpose is to show that forsome ~ € V either v(~,À (~),•) = ~ or

* * v(~,À (~),•) s. To ~his end, we first prove that À (~) depends continuous-

ly on This will be done as fellows. First we assert that for every ~ E V

and every À c [À ,À*(h)) a -

(7. 3. 7) g(t) < v(~1 À,t) < f(t) (t E [ 0 ,a]) •

In order to establish (7.3.7) we consider the function v(~,·,tl. Let

Àl ~ À2 . For all ti {0,~ .~ 1

, ••• ,~ } it then fellows that v(h,Ä 1,tl ~ n n+ n+r -# v(~,À2 ,t), since otherwise by i) of (7,3.4) and Theerem 7.2.6 we would

have v(~,À 1 ,·l = v(~,À 2 ,•J, which in view of ii) of (7.3.4) violatas the

assumption Àl # À2

• Because of the continuity of v(~,·,t), for every ~ E V

and every ti {O,~n'~n+ 1 , ••• ,~n+r} the function v(~,·,t) is strictly

monotonic (cf. Figure 1, where Àa < Àl < À2l.

v(h,X •• ·I

Figure 1.

In view of the strict monotonicity of v(~,·,t) it fellows that for all

ti {0,~ ,~ +l'''''~ +} and all À E [À ,À*(h)) n n nr a -

(t € [O,a]l •

From this and the inequality

(7 .3.8) g(t) s v(~,À*(.!!) ,t) s f(t) (t E [O,a]l ,

assertien (7.3.7) easily follows.

Now let (~k)~ be a sequence in V with li~~ ~ = ~0 1 it has to be shown

* * that li~~ À(~)= À (~0). As a consequence of Lemma 7.2.7 the sequence

165

(À*(~k)) is bounded, soit suffices to prove that every convergent sub­

sequence of (À*(~k)) has limit À*(~0 ). Consequently, we may assume, without

loss of generali ty, that li~-+<x> À* (~k) = 5: for some 5: E JR. In view of the

continuity of v(•,·,t) it follows that

lim v(~k'À*(~k) ,t) = v(~0 ,5:,t) k-+<x>

g(t) s v(~0 ,5:,t) s f(t) (t E [O,a]) .

Hence 5: E Ah and therefore (cf. (7.3.6)) 5: ~ À*(~0 ). If 5: < À*(~0 ) then _o

by (7.3.7) one would have

(7.3.9) g(t) < v(~o,5: ,t) < f(t) (t E [ 0 ,a]) .

On the other hand, there exists a convergent sequence (tk)7 with tk E (O,a)

* (k EN) and having the property that either v(~k'À (~k) ,tk) = g(tk) or

* v(~k'À (~k) ,tk) = f(tk). This, however, contradiets (7.3.9) as is easily

seen by letting k tend to infinity. Hence 5: = À*(~0 ), proving the continuity

of À* as a function of h.

As a next step in the construction of~ and s, a sequence m0 (~) ,m1 (~) , ..

. . , mr (~) , corresponding to an arbi trary ,~ E V, is defined by

- - n+~- n ~

[

min{f(t) -v(h,À * (h) ,t) lt; . 1 st sI; +'} (i=r,r-2, ... ),

(7. 3 .10) m. (h) := ~- * I min{v(h,À (h),t)-g(t) I; .

1stst; .}

- - n+~- n+~ (i=r-1,r-3, •.. )

We note that the nonnegative numbers mi(~) depend continuously on ~· The

existence of the required function ~ (s) will now be ensured if we prove

that an hE V exists such that mi(~) 0 for all i= 0,1, ••• ,r. Such an

h E V will be obtained by showing that the assumption L~=O mi(~) > 0 leads

toa contradiction. So, let L~=O mi(~) > 0 for all hE V. The vector­

valued function ~ defined by

(7. 3.11) F(h) := ___ a __ r

L i=O

m. (h) ~-

(h E V)

then is a continuous function from V into V. According to Brauwer's fixed

point theorem (cf. Kuga [29]) I a point n* exists with F(h*) = h*. Since

for every h E V there is at least one i0

E

* * one has hio = 0. Hence (cf. (7.3.5)) l';n+i -* * * * * 0

{0,1, ... ,r}

* t;n+i0-1

such that mi0 (~)

0. 'By (7,3.10) and

because v(~ ,À (~) ,l;n+io) = s(l';n+io) it now

This, however, contradiets the fact that

* follows that mi0 <~) > 0.

0,

166

0 __ .:;a __ m. (h*) r J.o-L m. (h*)

i=O l. -

We conclude that an ~ E V exists such that m1 (~) = 0 (i= O,l, ••• ,r). The n+r

corresponding sequence (1.: 1 } 0 evidently satisfies l,;n+i- ~n+i-l > 0

(i= O,l, ••• ,r) and the corresponding function v(~,À*(~},•) oscillates r

* times between f and g on [O,a]. Finally, if ris even (odd) v(~,À (Ë) ,•)

has a set of r + 1 oscillation points of type II (I) • ~ * * Consequently, we may takes= v(~,À (Ë),•) if ris even and ~ = v(~,À (~),•)

* * . if ris odd. If À (~) is defined by À (~) := inf Ah (cf. (7.3.6}), then l.n ...; -a similar way the existence of the remaining s (or ~) can be established.

(e:) We preeeed by showing the uniqueness of~· say. Let s 1 E Pr a be another ,_ function oscillating r times between f and g, and having a set of r + 1

oscillation points t 1 < t 2 < ••• <tr+l of type I. In order to prove that

s 1 = ~ it suffices to show (cf. Theerem 7,2.6) that s 1 and ~ coincide in

at least r + 1 points in (O,a], counting multiplicities.

Let ~ have a set of r + 1 oscillation points t l < t 2 < ••• < T ~+ 1 of type I.

Without loss of generality we may assume that for some i one has t 1 < ti,

s 1 (ti) = g(ti) < ~(Til and s 1 (til > ~(til = g(til. consequently, s 1 and ~

coincide in at least one point in (t1,til. Furthermore, s 1 and ~ coincide

in at least one point in each of the intervals [tj,Tj+l] {j = 1,2, ••• ,i-1)

and [tj,tj+1J (j = i,i+l, ••• ,r). Hence, s 1 and~ coincide in at least i-1

points in [ t 1

, ti] and they coincide in at least r- i + 1 points in

[t~rt' 1J, counting multiplicities. We conclude that s 1 and s coincide in

l. r+ -at least (i-1) +l+(r-1+1) = r+l points in [t 1 ,t~+l] and therefore by

Theerem 7.2.6 This establishes the theorem. 0

(e:) From this representation theerem for the set P (p ), we now easily obtain r,a n an analogous result for the set P (p ) by lettÎng e: tend to zero.

r •5!. n

THEOREM 7.3.5. Let g E~n. a> O, and Zet f E C([O,a])• gE C([O,a]) be

suah that an s € P (p) e:x:ists satisfying g(t) < s(t) < f(t) (t € [O,a]). r,a n Then funations s " P- (p l and '; € P {p l e:x:ist osaillating r times - r,a n r,a n . between f and g on [O,a], and suah that-! Îs) has a set of r + 1 osoillation

points of type I (II).

+ PROOF. Since Te:s ~ s {e: ~ 0) uniformly on every bounded interval of ~O

(cf. the proof of Theerem 7.2.9), it fellows that for sufficiently small

167

E > 0 the function T s satisfies the inequality g(t) < T s(t) < f(t) E . E

(t E [O,a]}. Let a. (E) be defined by a. (E:) :"' (d/dt) ~ T s (t) I t 0 ~ ~ E: =

{i 0,1, ••• ,n-1). According to Theerem 7.3.4 functions z8

E P~~1(pn) and

E P(E:) (p l exist oscillating r times between f and g on [O,a], and such r,:;& n

that (~ ) has a set of r + 1 oscillation points of type I (II) • By E:

letcing E: tend to zero we obtain the desired functions ~ and ~. 0

Let~= (cx 0 ,cx 1 , ••• ,cxn- 1)T E JRn, andlet the set Fcx(J) (J =JR~ or J

with a > 0) be defined by

[O,a]

{7 .4.1) Fcx(J) := {f E w(n) (J) I llf!IJ ~ 1, f(i) (0) = Cli' i= O,l, ... ,n-1}

In this section we shall be concerned with the problem of minimizing + llpn (D) f!l+ on Fa (lR0l, where pn (D) is a given differential operator. It

will be proved that a perfect !-spline f with, in genera!, infinitely many

knots solves the problem. The extremal function will be obtained as the

limit of a sequence of perfect !-splines (s )00

1 with s E P (p ) , and

r - r r,~ n each sr is an extrema! function for the problem of minimizing llpn{DlfiiJ on

(J), where J is an appropriate finite interval depending on r.

In order to ensure that Fa {lR~) is nonempty the following lemma is of

importance.

LEMMA 7.4.1. The set F {lR~) is nonempty if and only if the polynomial

p 1

E ~ 1, determine~ by the aonditions p(i)1 (0) =ex. (i"' O,l, ••• ,n-1}, n- n- n- ~

has the property that 1Pn_1

(t) I ~ 1 in a right-neighbourhood of O.

PROOF. If a function f E Fa(lR;) exists, then according to Taylor's formula ~ n n

one has pn_ 1 (t) = f(t) + 0(t ) (t + 0). Hence lpn_ 1 (t) I s 1 + 0(t ) (t + 0).

Since pn-l E ~n- 1 we may conclude that lpn_ 1 (t) I s 1 in a right-neighbour­

hood of 0.

Conversely, let lp 1 (t) I 5 1 in [O,t0J for some t > 0. If lp 1 (t) I n- 0 n-

for all tE [O,t0],then p 1 <tl 1 or p 1 (t) = -1 and we may choose n- n-f(t) = 1 or f{t) = -1 (t ElR;). If there exists a t 1 E [O;t0) such that

+ lpn_ 1 {t1) I < 1, then a function f E Fcx {lR0) can be obtained as fellows.

Let q2n-l E ~2n_ 1 be the polynomial ;atisfying

168

(i= 1,2, ••• ,n-1) ,

q2n-1 (OJ = 1 ' q2n-1 (l) = 0 •

The function fÀ defined by

(Û ~ t < t1) 1

(t- t1) f, (t) := p (t)q --

/\ n-1 2n-1 À .

0

If a function g € Fa(m~) exists such that pn{D)g{t) = 0 (t > 0), then, of

course, the minimum-of llpn{D}ftl+ on Fa(m~l is zero and gis an extrema! - = function. If such a function g does not exist then a sequence (srl_ 1 of

perfect .C-splines € P r (pn) will be determined" converging to an extrema!

function for the problem stated in the beginning of this section. As will

be evident from what fellows, this approach makes essentially use of the

assumption that

(7 .4. 2} P (p l n F (IR +0

> = 11 r,g n ~

(r € :Nol •

The condition pn(O) = 0 is sufficient to ensure that (7.4.2} holds,as then + for every r € :N

0 any function s € Pr(p

0) (r ~ 0) is unbounded on m

0• For

this reason we shall assume from now on that the following conditions are

satisfied:

(7 .4.3) F~(m~) rf !J, Ker(pnl n F~(m~l = !;J, p0

(0) = 0 •

We note that {7.4.3) implies that la0 1 ~ 1.

By the continuity of the perfect .C-splines and by {7.4.3), it fellows that

for each s € Pr 1~{p0 } with r ~ -1 there exists a positive number ps defined

by

(7 .4.4) ps :=max{p lls(t)l ~ 1, O~t~p}.

169

DEFINITION 7.4.2. FOT' r+1 E :t!O Wedefine

(7 .4.5) A r

sup {p I s E P (p ) } s r,~ n

In what fellows it will be shown that for each Ar the problem of determining

a function f E ([O,A ]) that minimizes lip (D)fii[O A] on F ([O,A ]) is r n 1 r ~ r

solved by an appropriate sr E P (p ) exhibiting a specific oscillation r,~ n

behaviour.

n THEOREM 7.4.3. Let r+1 E :t~0, and let~ E E and the monie polynomial

pn E ~n be sueh that (7.4.3) is satisfied. Then there exists a perfect

.C-spline sr E P (p ) oseiUating r + 1 times beween -1 and 1 on (O,A J. r ,a n r Moreover, any s Z P (p } having this oseillation property satisfies r r,~ n

(7 .4.6} liP (D}s 11 =min {lip (D)fllco J I f E Fa([O, n r n ,Ar ) } .

PROOF. Let ó > 0. Since a function sE P (p) exists with ls(t) I $ 1 on r ,a n [O,A ] , it fellows that I (t) I < 1 + ö on-[O,A ] for sufficiently srnall

r r c > 0, where T

8 is the operator defined by (7.2.5). Taking

ai (c) .·= (aat)i (t) I t=O (i= O,l, ••• ,n-1)

and applying Theorern

s E P (c) ( } (p ) and ~e: r,Cl e: n

7.3.4, we obtain two e:-approxirnate perfect t-splines,

€ P (€:) ( l (p ) 1 oscillating r times between -1 and r,~ c n

1 on [O,Ar]. Letting ö and c tend to zero we rnay assume, in view of

Theerem 7.2.9, that ~~ converges tos <:: P (p ) and converges to ,__.(;,. - r,a. n ; EP (p ), where both s and; oscillate; times between -1 and 1 on r,a n ~

[O,A ].-We assert that s(A) ;(A). If s(A) ~;(A) then for sufficient-r ~ r r ~ r r ly small e: > 0 one has that (A ) ~ ; (A ) • In view of their oscillation

r E: r ~

properties (cf. Theorern 7.3.4) the functions ~e: and coincide at precisely

n+r points, so Lemma 7.2.10 rnay be applied with t 0 replaced by Ar. Thee:-

approximate perfect

Consequently, s (A ) = ~ € r

obtained in this way is denoted by se.

(A ) + (A ) ) and Is (t) I $ 1 + ö (t E [O,Ar]). r r E

Letting again ö and E tend to zero we get a function s E P (p ) with the r,~ n

properties

I s<tl I $ 1 s (A ) = ! (s (A ) +;(A ) ) • r ~ r r

From the definition of Ar it fellows that I~(Ar} I = I~(Ar) I ls!A >I r 1,

170

contradicting the assumption that s(A) ~ s(A }. Hence, either s{A) D - r r - r

= s(Ar} = 1 or ~(Ar) s(Ar) = -1. These equalities imply that either! or

s oscillates r + 1 times between -1 and 1 on [ 0 ,A ] • Consequently, P (p ) r r,~ n contains at least one function sr that oscillates r + 1 times between -1

and 1 on [O,Ar]. At this junction it is necessary to distinguish between

the cases la01 < 1 and la01 = 1. If la01 '< 1 then, evidently, the function

sr oscillates r + 1 times on (O,Ar]. If, however, la.0 1 = 1 then we proceed

as fellows. Without loss of generality we aasurne that a. 0 1. Since (7.4.3)

holds one has (a. 1 ,a. 2 , ••• ,a.n-l) # (0,0, ••• ,0), and if forsome

jE {1,2, ••• ,n-1} a 1 = a. 2 = ••• = a.j_1 0 and aj # 0 then a.j < 0. It

fellows from the reasoning above that for every 6 > 0 an s(~) E P (p ) o r,a. n

exists oscillating r + 1 times between - 1 - ö and 1 + ö on an intervaÏ con-

taining [O,Ar]. Hence, in view of Lemma 7.2.8,. there is a sequence {6k>7

of positive numbers with li~400 ök = 0 such that the corresponding sequence

of perfect .C-splines (s (o ) ) converges to an s E P (p ) , uniformly on k r,~ n

[O,Ar]. It may also be assumed that s (ok) has a set of r + 2 oscillation

points 0 < Tl,k < T2,k < ••• < Tr+2,k sAr such that Ti := li~ Ti,k

(i = 1,2, ••• ,r+2} exist and, moreover, that all sets of oscillation points

are of the same type. If T1 = 0 then, because of s(t 1l = s(O) = a. 0 = 1, the

sets of oscillation points are of type II (cf. p. 162). Hence s(6k) (T 1 k} = (1) . (j) ,

= 1 +ok. Since scok} {0} = 1, s(Ök) {0) = 0 Ci=1,2, ... ,J-l), s(Ök) {0) =aj "0,

it fellows that

As ök > 0 we conclude that a.j > 0, contrary to our observation above and

consequently T 1 > 0. It fellows that a lso in the case I a.0 I = 1 there exists

an s t: P (p ) oscillating r + 1 times between -1 and 1 on (O,A J. r r,~ n . r In order to prove the second part of the theerem we note that every

f t: F ([O,A ]) coincides with s in at least n + r + 1 points, counting ~ r r

multiplicities. Applying Theerem 7.2.3 wethen obtain {7,4.6). This com-

pletely proves the theorem.

As a simple illustration of Theerem 7.4.3 we consider two examples.

0

EXAMPLE 1. Let r 01 fl T 3

(1,0,-1) E lR and p3

(D) D(D2-I).

0

-- t

-1

Figure 1.

A sketch of the extrema! function s0

E P01 ~(p 3 ) is given in Figure 1.

Taking into account (7.2.1) we have

s0

(t) = 2 - cosh (t) + c (sinh (t) - t) (t E lR) ,

where the constant c is determined by the condition that s0

has minimum

value -1 fort~ 0. It fellows that c = 1.044 and A0

= 5.327. Hence, if

f E Fg_([O,A0 ]) then, according to (7.4.6),

llf'" -f'll ~lis"' -s'll = 1.044 [O,A0J 0 0

171

Figure 1 also contains a sketch of s_1

E P_ 1 ,g_(p3), i.e., s_ 1 E Ker(p3l

satisfying the conditions s_ 1 (0) = 1, s~ 1 (0) = 0 and s:1

(0) = -1. Obviously,

s_1

(t) = 2 - cosh(t) (t E lR).

EXAMPLE 2. Let r 1, g_

0

-1

Figure 2.

172

The extrema! functions s_1

, s0

and s1

(cf. Figure 2} are given by

{t € lR) ,

x1 = 4 being the only knot of s 1 • Obviously, A_ 1 = 1, AO = 2 + 212 and

A1 = 6 + 212. The perfect spline s 1 E Pl,·a (p2) oscillates twice between -1

and 1 on (O,A1J and therefore, by Theerem 7.4.3, llsïll minimizes llf"II[O,Al]

on Fa([O,A1Jl. Note that the tunetion t

0 defined by

(0 ~ t ~ 'C 1} ,

(t>-rl},

also minimizes llf"II[O,Al] on Fa{[O,A1]), so s 1 is not uniquely determined

by i ts extremal property. However, as a consequence of the following

theorem, every f € Fa([O,A1J> satisfying (7,4,6) coincides with s 1 on

[O,-r1].

THEOREM 7.4.4. Let r € ~o~ Zet~= (a0

,a 1 , ••• ,an_1lT E lRn (n ~ 2), and Zet

s € P (p ) eatis fY the fo UOüJing wo condi ti ons: r,~ n

i) the'l'e e:ciet r + 1 points 0 < -r 1 < -r 2 < ••• < 'r+l euah that

(7.4.7) (i= 1,2, ••• ,r+l)

ii) s has exaatty r knots 0 < x1 < x2 < ••• < xr eatiefying the inequaZi­

ties

{7.4.8) (i= 1,2, ••• ,r+l) ,

~he'J.'e xj :=~in aaee j > r; mopeover, (7.4.8} is void if r o.

Then for aU f E F ~ ([O,tr+l ])

{7.4.9)

~ith equaZity if and onZy if f(t) = s(t) (tE [0,-rr+l]).

PROOF. Using Taylor's formula and an auxiliary function a, we first derive

arelation valid for any function in AC(n} ([O,•r+l]). To this end, let~ be

the fundamental function corresponding to the differential operator pn(D},

and for t € lR let H be defined by the determinant

(7,4,10) H(t)

(j) + (T 1 - t)

<p+('l xl)

(j)+(T 2 - t)

~p+(T2 xl)

Obviously, H can be written in the form

( 7. 4 .11) H ( t) r+1 I B rp (T - t)

k=1 k + k

where (cf. p. 4 for notation)

(t E lR) 1

~-1

It follows from Lemma 1.4.19 and (7.4.8) that

(7,4.12) sgn(f\) = (-l)k-l (k 1, 2, ... 1 r+l)

<p+('r+1-t)

(jl+('r+1-x1)

!p (T - ') + r+l

x r

173

Taking into account the definition of rp+ (cf. p. 6) and (7,4.10) one easily ( . )

observes that H J (T 1) = 0 (j = 0,1, •.• ,n-2) and H(:x:i) = 0 (i= 1 1 21 ... ,r). r+ . We preeeed by proving that H changes sign at the points :x:i. If tE (:x:k,xk+l)

(k = 1 , 2, • , • , r-1) then

H(t)

and Lemma 1.4.19 again yields

k sgn(H(t)) = (-1) (~ < t < ~+1

In a similar way it may be shown that

sgn (H (t)) = { 1

(-1) r

t ~+1

lil (T - •) + r+1

x r

k .1 I 2 1 • o o I r-1 ) •

(n) Now let f E AC ([0,Tr+

1Jl. Then by Taylor's formula (cf. Lemma 1.4.4)

and ( 7 • 4 • 11)

174

r+1 'r+l

(7.4.13) H !nl pn !Dl f !nl dn I 1\ k=l f

0

where the coefficients y 0 ,y 1, ••• ,yn_1 do notdepend on f.

Consequently, for every f E F ([0,-r 1Jl

~ r+

(7.4.14)

where a.0 ,a 1 , ••• ,a.n·· 1 are

We now apply (7.4.13) to

and Bk (cf. (7 ,4,12) and

(7.4.15)

'r+1

f 0

lacnl ldnllp (Dlfllco J , n ,r r+l

the components of ~·

s E P (p l . In view of the sign structure of H r ,a n (7.4.7)), s satisfies the relation

'r+l

J !Hinl ldnllpn!DlsH

0

Hence, combining (7.4.14) and (7.4.15), we conclude that

and, moreover, that lip (Dlsll =liP (D)fii[O J if and only if f(t) = s(t) n n •'r+1

(0 !> t s; T r+l) • This proves the theorem. 0

We note that the existence of an s E P (p ) satisfying conditions i) and r,a. n ii) of Tbeorem 7.4.4 is not warranted. with respect to this the following

example is of interest.

EXAMPLE 3. Let p 3 (o) = o3 and let s 0 be an element of P0 (p3) (cf. Definition

7.2.1), the graphof which is given in Figure 3,

175

0

-1

Figure 3.

A perfect spline s 1 E P1 (p3) is now obtained from s 0 by adding an appro­

priately situated knot x1

such that s1

(t) = s 0 (t) (0 st s x1

J and such

that s 1 exhibi ts the oscillation behaviour as sketched in Figure 3. If we

take~= (s0 (0) ,s0CO) ,sÖ{O))T it fellows that s 1 E P 1 ,~cp 3 ). In fact, we

have {cf. Figure 3)

sl(t) = so(t) + 2-us"'ll(t-x )3

3 0 1 + (t E lR)

Hence llsi"li = 11 "'11. Since x 1 > -r 1 , condition (7.4.8) does not hold and so

Theerem 7.4.4, with s replaced by s 1 E Pl,a(p3), does not apply. However,

in this specific case, if f E F~([O,A 1 ]) s;tisfies llf"' 11[0,Ao] =lis i" 11 = lls0"11,

then Theerem 7.4.4 may be applied with s replaced by s0

E PO,a(p3), and it

fellows that f coincides with s0

on [O,-r 1J. -

EXAMPLE 4. Let r 5, ~ = (0,0,0,6) E JR4 and p 4 (D) = o4 • We note that a

and p 4 are chosen such that condition (7.4.3) is satisfied.

-t

-1

Figure 4.

176

In order to obtain the perfect spline sS of Theerem 7.4.3, one has to solve

a system of nonlinear equations of the form (cf. (7.2.1))

[

SS{'ti) = T~-c5r~-2c5 jt{-l)j(Ti -xj): = (-1)1+1

s'(-r) = 3t2 -4c t_3-8c I (-l)j(t -x.) 3 = 0 s i i 5 i 5 j=1 i J +

(i= 1, ••• ,6) ,

(i= 1, ••• ,6) ,

in the twelve unknowns r 1,r 2, ••• ,T6 ,x1,x2, ••• ,xS,cS, where

cs = lls~ 4 l 11/24. This is done numerically by means of an iterative

method. The numerical values of the knots x1

,x2, ••• ,x5 and the oscillation

points r 1,r 2, ••• ,r6 are given in Table 11 cs = 0.4786S.

x1 1.15913 Tl 1.65664

x2 2.S6369 't2 3.3246S

x3 4.13349 T3 4,93522

x4 5.73972 '4 6.S4299

x5 7.35339 Ts 8.14382

T6 9.58318

Table 1

Note that the knots x1,x2, ••• ,x5

and the oscillation points r 1,T 2, ••• ,T6 satisfy the inequalities (7.4.8). Since s

5C-r 1) = 1 and s~4 l (0+) =

=- 11.4876 < 0, condition (7.4.7} is also satisfied and Theerem 7.4.4 may (4)

be applied. Consequently, if f0 <:: F!!.([O,T6JJ minimizes llf II[O,T6

] on

F~([O,r6 J> then f 0 (t) = s 5 (t) cos t s T6).

On the foregoing pages we have shown that a perfect t-spline s <:: P (p ) r r ,a. n

oscillating r + 1 times between -1 and 1 on (O,A ] solves the problem of r

minimizing lip (D) fii[O A ] on F {[O,A ]) • Now we want to prove that a con-n •r a. r

vergent subsequence of {s ,~1 solves the corresponding problem of mini­r -

mi zing llpn (D) fll+ on Fa (lR~). For this purpose we need the following

proparty of the sequence {A >=1• r -

LEMMA 7.4.5. Let r+l <:: JN0

• and Zet 5:. <:: :rRn (n ~ 2) and the monic polynomial

p <:: w be euah that {7.4.3) ie eatis~ed. Then the eequence (A l=1 ie n n r-nondeareaeing and limr~ Ar = =.

177

PROOF. It immediately fellows from the definition of Ar (cf. (7.4.5)) that

(Ar) : 1 is nondecreasing. Hence limr+w Ar= ~ with ~ s ~. Now let (sr):l be

a sequence of perfect !. -splines E P (p ) oscillating r + l times r ,:::_ n

between -1 and 1 on (O,A ] and r

7.4.3). Since sr2 E Fa ([0,Ar1Jl

is nondecreasing. Moreover, it

possessing property (7.4.6) (cf. Theerem

for all r2

?: r1

, the sequence <llpn(D)srlll: 1 is bounded since llp (D)s 11 lip (D)fll for n r n +

+ all f E (lR0). If ~ < "" then the boundedness of ( jjpn (D) srlll tagether

with the inequality lsr(t) I $ 1 (rE N0 ~ tE [O,A0]) would yield that

(IJs~II[O,~]) is bounded. This, however, is incompatible with the number of

oscillations of s on [O,~] tending to infinity as r ~ ""· Hence ~ = oo 0 r

THEOREM 7.4.6. Let a E mn (n?: 2) and the monia poZynomiaZ p E ~ be suah - n n

that (7.4.3) is satisfied. Then there exists a perfeat f.-spZine s, corre-

sponding to the operator pn(D), with infinitely many knots and having the

extremal property

(7.4.16) llpn(D)sll

PROOF. Let s E P (p ) be a perfect !. -spline oscillating r + 1 times r r,a n

between -1 and 1 on-(O,A] and satisfying (7.4.6). First we show that the r . total number of knots of sr in a bounded interval [O,N] is bounded as

r ~"'·Let Nr denote the number of knots of sr in [O,N]. Since by Lemma

7.4.5 Ar~ oo (r ~ ~>, the open intervals (N,Ar) are nonempty for suffi­

ciently large r. On {N,Ar) the number of knots is bounded by r Nr' and so

by Lemma 7.2.4 cannot have more than r-Nr+n zerosin (N,Ar). Since

sr oscillates r + 1 times between -1 and 1 on (0 ,Ar], of which, say, Kr

times on (0 ,N) 1 one has r- Kr r- Nr + n. Hence Nr s n +Kr. Now the

sequence {Kr):1 is bounded, since otherwise the number of oscillations of

on (O,N) would tend to infinity as r..,."' and so Us~II[O,N] ..,. "' (r..,. ~J,

which (cf. the proof of Lemma 7.4.5) cannot occur. Consequently, a number

qN < "' exists such that every sr coincides on [O,N] with a function in

PqN,:::_(Pnl. In view of the compactness property (cf. Lemma 7.2.8) there is

a subsequence of (sr) converging to an s E Pq a<P ) I uniformly on [O,N]. N'- n

By a standard diagonal procedure one obtains a subsequence (sr.l of (s ) J r

such that srj..,. s (j..,. oo), uniformly on every bounded interval of m~. Moreover, lip (Dlsr.ll -rlip (D)sll (r. ~ ""), and sis a perfect f.-spline with

n J n J infinitely many knots, but with finitely many in every bounded interval.

178

In order to

l!pn(D)fll+ <

llpn {Dl fll+ <

proparty of

+ prove (7.4.16), let f € Fa(JR0) be a function such that

lip (D}sll. Then fora suffÎciently large j one would have n

lip (Dlsr.ll· However, this inequality contradiets the extramal n J

sr· as given in {7.4.6). This proves the theorem. J

In the preceding sectien the problem of minimizing llpn (D) fll + on Fa (JR~) is

considered, Here we explain i ts re lation to the set ar: {p0

) •. To illustrate

this two examples will be given. We reeall that pn E 11n has only real

zeros and that pn{O) = 0,

Let the function M be defined by (cf. (7.4.16))

(7.5.1)

in case Fa (JR~) = ~ we put M (~) := "'.

n (~ € l< )

Obviously-;- the function M is convex and therefore (cf. Roberts and Varberg

[49, p. 93]) continuous on the open set

(7,5,2)

In view of the definition of r+(p) (cf. {5,3.1) and (5.1.3)) a point m n

a E JR0 belengs to r+(p) if a function f E F (p ,JR0+) exists, i.e., a

- m n m n function f E AC (n) (JR~) for which llfll+ :s: m and lip (D) fll+ ~ 1, such that

(j) . + n f (0) ~ aj (J "'O,l, ••• ,n-1). If f E F (p ,JR

0) then, evidently,

m n + f/m E F1 (p ,JR+0l and it fellows that M(a/ml ~ 1/m. Hence, a Er (p) if n - - m n and only if M(~m) s 1/m. The following lemma characterizes the boundary

points a of r+(p) in termsof M<~m). - m n

LEMMA 7.5.1. Let m > 0 and let~ • (a0

,a1, ••• ,a0

_ 1)T E JR0 ~ith la0 1 < 1.

Then a E ar+(p l if and only ifM<a!m> "'1/m. - m n -

+ + PROOF, If a € r (p ) then M{a/m) :S: 1/m. Now let a E ar (p ) and assume - mn - - mn that M(~m) < 1/m. Then by the continuity of M there is some À > 1 such

+ that 1Àa0 1 < 1 and M(À~ml < 1/m. This implies that À~ E rm(p0), contra-

+ dictory to the assumption that a E ar (p ). Consequently, M(a/m) = 1/m. - m n -

D

179

Now let M<~m) 1/m. Then, obviously, a E r+(p). If a i ar+(p ) then - m n - m n

some À> exists such that 1Àa0

1 < 1 and Àa E r+(p). Hence M(a/m) ~ -1 -1 m n -

~ (Àm) < m , yielding a contradiction. 0

Lemma 7.5.1 tagether with Theerem 7.4.6 gives

THEOREM 7 5 2 L t 0 l t ( ) T JRn . .,_, , I ..• e m > , e .::. = a 01 a 1 , ... ,an_ 1 E W~vn 1a0 < m,

and let pn E rrn be a monio polynomial having only Peal zeros with

pn(O) = 0, The? .c:.. E ar;(pn) if and only if a perfeat f.-spline sE Fm(pn,JR;)

exists with s(~) (0) =a. (i O,l, ... ,n-1) and sueh that ~

(7. 5. 3) llpn(D)sll =

min {llpn(D)fll+ I llfll+ ~ m, f(i) {0) =ai, i 0, 1, .•• 1 n-1}

REMARK. In case la0 1 =m it is possible that the minimum in (7.5.3) is

smaller than one; it may even be zero as is illustrated by the simple T + 2 T example: (1,0) E ar1 (D) and M((1,0) ) = o.

1 •

Theerem 7.5.2 and Lemma 7.5.1 enable us to describe the boundary of r+(D 3) . m

explicitly by deterrnining various extrema! functions (cf. Exarnple 1 below).

To do the sarne for ar+(D4 ) is much more difficult, as is evident from m·

Exarnple 2. For the construction of even one single point of ar+(D4) rather m

elaborate numerical computations are needed.

EXAMPLE 1. Befare descrihing the set ar+(D3) a preliminary remark is in 3 + -1 m 1/3 3 + order. If f E Fm(D , JR0) then tl-) m f(m t) belengs to F1 (D 1 JR0).

. T + 3 -1 -2/3 -1/3 Consequently, ~f!:.. = (a0

,a1,a

2) E arm(D) then (m a

0,m a

1,m a

2) E

E 3f:(D3). Hence it suffices to describe ar:(D

3).

On the interval [O,h] the perfect Euler spline f(D 3 ;h,•) is given by (cf.

p. 131)

(7.5.4) 3 E(D ;h,t)

We reeall {cf. (5.5.1)) that 3 3

11 E (D ; h, •) 11 = h / 24 ( cf. p. 3 accordingly 11 E (D ;h, •) 11 = 1.

f(D\h,t+h) =- E(D3 ;h,t) (tE lR) and that

131}. In what fellows h is taken 2~ and

180

Now for any n E [-!h,O] and r+l E ~O a perfect l-spline sr+l,n is con­

structed as fellows. Let § E P (03) be the perfect l-spline coinciding 3

r r with- E(D 1h,•) on [O,rh] and having the knots h,2h, ••• ,rh (cf. Figure 1,

where r 3).

-1

Figure 1.

Associated with §r we consider the perfect l-spline sr+l with knots

n,h, ••• ,rh and coinciding with §r fort~ n (cf. Figure 1}. Letting

-t

y E [-h,-!hJ be such that s 1 (y) = -1, wedefine (cf. Figure 2) s +l by r+ r ,n the translation sr+l, I) (t) sr+l (t + y) (t € lR).

-1

Fiqure 2.

Taking into account Figure 2 it is evident that

and s 1

oscillates r + 2 times between -1 and r+ ,n

7.4.3 it follows that llf"'ll ~lis'" 11 = 1 for + r+l,n

-t

3 in general sr+l,n E Pr(D } ,

1 on (O,Ar+l]. By Theerem

any f € Fa([O,Ar+l]), where

g, := (sr+l (0) ,s'+l (0) ,s"+l (0)) T = (s1

(0) ,s1• (0) ,s•

1• (0)) T •

,n r ,n r ,n ,n ,n ,n

Letting r tend to infinity and using Theorem 7.5.2 we conclude that

(s 1 (0) ,s1' (0) ,s•1• (O))T € ar+1 co3). Using similar arguments we also infer ,n ,n ,n

that

(7.5.5) (sl (T) ,st' (T) ,s't' ('t')) T € ar+ (o3) ,n ,n ,n 1

(T € [O,n-y]) •

Other boundary points of r;co3) can be found by using the following

181

construction, Lets* be the perfect spline with knots h,2h, ••• and coin­

ciding with -1 + h3/24- E(D\h,t) fort~ 0 (cf. Figure 3); here

h E (0,2'03),

-t

-1

Figure 3.

In view of (7.5.4) * s coincides on [-!h,h] with the polynomial q 3 given by

1 q3 (t) := 6 ( t ( JR) •

One easily verifies that q3

oscillates once between -1 and 1 on any interval

of the farm [T,d], with TE [-!h,h] and d being detern:.ined by q3

(d) 1 (cf.

Figure 3). Since can be interpreted as a perfect spline having zero

knots, it fellows from Theerem 7.4.4 that llf'"ll ~ llq'"ll T [O,d--r] 3 f E Fa ([O,d--r]) with ~ := (q

3 (T) ,q:} (T) ,q3 (<)) • Hence l!f'"ll+ ~

1 for all

for all - + * f E F!:!. ( m

0J • Since s E F~ (lR6) , Lemma 7. 5.1 applies and so

(7 .5.6) ( -!h "; ' "; h) •

A sketch of r~ (D3) in x y- z space can now be given; this is done in

Figure 4. Because of the symmetry of r;(o3) it suffices to consider the

half-space y ~ 0 only.

The curves and 2 in Figure 4 consist of points as given by (7.5.6) for

two different values of h E (0,2\73). Similarly, the curves 3 and 4 consist

of points as given by (7.5.5) for two different values of n E [-~,0].

182

x

I I I I I I

z

: ,. ... ""' /\ ~----1-\----

.... ..-" I I \ /".. I I \

/ I I \

Figure 4.

-1

y

4 T 4 EXAMPLE 2. Let p 4 (D) = D and let~= (O,O,O,a 3} €. m. • The problem we pose

+ 4 is todetermine a 3 such that ~ e. arm(D }. As pointed out, we may assume

without loss of generality that m = 1. According to Lemma 7.5.1 the point T + 4 +

(O,O,O,a3) € ar 1 (D ) if and only if M<O,O,O,a3) = 1. If f €. F.(O 0 0 a } (lRo) 1/3 + f f I 3

then (t,.....f( (6/a 3) t)) r.;: F(O 0 0

G) (lR0

) and it fellows that the equality I I I 4/3

M(OIO,O,a3) = 1 is equivalent to M(0,0,0 1 6) .. (6/a3) • So, in order to

determine a3

, it suffices to compute M(0 1 0 1 0 1 6}. Similar to the proef of

Theerem 7.4.6 this will be done by determining a sequence of perfect "' . 4 T splines (s } 1 Wl.th s r.:: P (D ) , a = (OI01016) 1 and such that sr

r - r r,a -oscillates r + 1 times between -1 and 1 on (O,A ]. Hence, by setting (cf.

r Example 41 p. 175)

(7,5,7) (t €. m.) ,

we aim to compute oscillation points 0 < Tl,r < T2 ,r < ••• < Tr+l,r such

that

183

(7 .5.8) S (T. ) = (-1)i+l, r ~,r

I (T • ) ~,r

0 (i 1,2, •.• ,r+ll.

In view of (7.5.8) the knots x 1,r,x2,r, ••• ,xr,r' the oscillation points

Tl,r'T 2,r'''''Tr+1,r and the constant er> 0 are determined by a system of

2r + 2 nonlinear equations. For r = 5 ,6, ••• , 19 numerical val ues for

x 1,r,x2,r, ••. ,xr,r' T1,r,T2 ,r'''''Tr+l,r' er are obtained by means of an

iterative method. Being interested in M(0,0,0,6) we only give here the

numerical values of lls;4

)11 (r = 5,6, ••• ,19).

r lis (4) 11 r 11 s (4) 11 r lis (4) 11 r r r

5 11.48760 10 11.48773 15 11.48785

6 11.48764 11 11.48773 16 11.48787

7 11.48764 12 11.48774 17 11.48789

8 11.48769 13 11.48780 18 11.48789

9 11.48771 14 11.48783 19 11.48792 ..

Table 1.

(4) 19 We abserve that the sequence ( 11 s r 11)

5 is monotonically nondecreasing as

it should be, sinces minimizes llf(4ln[O A] on F ([O,A ]), s +l miniruizes (4 ) r , r ~ r r

llf II[O,Ar+l] on Fa.([O,Ar+l]) and (cf. (7.4.4) and (7.4.5)) [0 c[O,Ar+l].

Taking into account the data of Table 1, ene may reasonably expect that

M(0,0,0,6) = 11.49 rounded to two digits. Hence a. 3 0.9614.

4 + REMARK. We note that an extrema! tunetion f E F 1 (D , JR

0) for which

(f(O) ,f' (0) ,f" (0) ,f'" (0)) = (0,0,0,0.9614) E ar+(o4) can easily be extended 4 1

to a function g E F 1 (D ,lR) by means of

g(t) ... f f(t) t- f(-t)

(t ;:" 0) ,

(t < 0)

consequently, ~ = (0,0,0,0,9614) E ar 1 (D4

) and gis an extrema! tunetion

for a related Landau problem on the full line.

184

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192

LIST OF SYMBOLS

Symbols are defined on the pages listed.

AC(n) (J) 3 F(pn,T) 132

A 169 F{P ,T) 136 r n

Bh(pn;') 29 F I) (J} 167

B n,h 29 Fp 110 n

C(J} 3 h 7

C (n) (J) 3 H 38 n,v

C(aO,al, •.• ,aN-1} 83 Kn 46

D 5 K 87 x,y

Dk -5 K 95 x,a

Di 8 K 96 X,~)

D[i] 13 K 90 x,a

E 44 Ker(pn} 5

EA (€) 105 Ker(pn (D)) 5

Ê (el 105 Ker(pn,T) 133

E 58 K(x0 ,x1, ••• ,xn+!) 45 n

f(pn;h, •) 66 L{n,k) 47

E(p2

;co, •} 116 Lco(J) 3

E + (p2 ;h, • l 120 L00

(lR,T) 132

!<to,tl, ••• ,tm-1> 4 M0 (f,al 12

f[x0 ,x1, ••• ,xn] 25 M( (Di)~ ,f,a) 13

f[pn;xO,x1, ••• ,xn] 25 M{pn;xO,xl, ••• ,xn;•) 28

Fm{pn,JJ 103 Mj(pn;•) 28

193

M n,j

29 + S (a 1 ,a2 , •.• ,an) 11

-M

n+1, (x) 42 s (f, (a,b)) 11

Mpn 110 T 155 E

M (E:) 178 Upn 135

N n,i

45 w(n) (J) 3

PC(J) 3 n

Z((Di)1,f,(a,b)) 14

PC(n) (J) 3 cp 6

pn 5 cp+ 6

* lcpo Pn 60 cp1

:m-11 4

Pn (D) 5 ,to t1 m-1

p* (D) 24 rm(pn,J,t;) 104 n

~

Pn 56 rm(pn) 110

P r(pn) 153 r+ ( l m pn 110

p (p ) 153 r (pn,T) 133 r,~ n

p (E) ( ) r pn 155 r (pn,T) 136

p (E) ( ) 155 4> 38 r,::_ pn n,\!

p 137 4> n+1, (x)

43 n

'11 5 lj!À (pn;h, 59 n

TI n-1

62 Ij! À 59

Qn 138 lj!E (pn;h, 67

s (pn,h) 7 '!'À (pn;h, 59

S (pn (D) ,h) 7 'I' À 59

-s (a1,a2, •.. ,an) 11

194

SUBJECT 1 NVEX

Numbers indicate the pages of the text on which the subjects are defined,

treated, etc.

B-splinas

basic proparty of

definition of

Fourier transferm of

hyperbalie

minimum supramum norm of polynomial

normalizad polynomial

polynomial

recurrence relations for

total positivity proparty of

trigonometrie

variatien diminishing proparty of

with equidistant knots

Budan-Fourier theerem

classical

for t -splines

generalized

Chebyshev system

definition of

extended

extended weak

weak

circulant matrix

coincident block

condition number of a normalized B-splina basis

differentiation formula

disconjugacy of a differentlal operator

31 - 32

28

38

29

44-54

45

29

38-44

33

29

33

43

11

24

17

19

19

19

19

83

4

45

74

8

divided difference

error estimates

for cardinal !-spline interpolation

for periodic cardinal !-spline interpolation

extremal properties

of perfect t-splines

of T-periodic perfect Euler t-splines

of T-periodic perfect !-splines

extremal function

full-line case

fundamental function

Green's function

half-line case

inequalities

of Landau

of Kolmogorov

interpolation

cardinal t-spline

error estimates for càrdinal t-spline

error estimates for periadie cardinal !-spline

fundamental solution of interpolation problem

Hermite

periodic t-spline

property of perfect l-splines

property of "t:-approximate" l-splines

intrinsic error

knots

definition

multiple

multiplicity of

simple

195

25

91 93

94,99,101

167

149

146

103

104

6

6

104

102

102,103

75

91 -93

94,99,101

78

4

75

154

155

105

5

5

5

6

196

.C-splines

Budan-Fourier theerem for 24

cardinal 7

definition of 5

degree of 5

Euler 58

exponential 57

normalized exponential 59

number of zeros of 23,24

order of 5

periodic cardinal 75

relation between finite differences and derivatives of cardinal 72

Landau problem

for periodic functions

for secend order differential

for secend order differential

for third order differential

for third order differential

mesh di stance

nodes

oscillation points

de fini ti on of

of type I

of type II

Peano's remainder formula

perfect t-splines

compactness properties of

definition of

extramal properties of

interpolation proparty of

one-sided perfect Euler

perfect Euler

representation theorems for

satisfying initial conditions

wi th knots in JR~ "e:-approxima te"

operators (full-line case)

operators (half-line case)

operators {full-line case)

operators (half-line case)

116-120

120-125

127-131

179-181

7

55

162

162

162

10

158

7

167

154

120

66

162,166

153

153

155

polynomial splines

B-splines

cubic

definition of

quintic

polynomials

Bernoulli

Euler

Euler-Frobenius

exponential

exponential !­

monic

normalized exponential

normalized exponential !-

power growth

representation theorems

shift operator

smeething operator

strict convexity

sign changes

streng

weak

strong zeros

definition of

multiplicity of

Taylor's formula

T-periodic function

truncated power basis

(uni) solvent family

197

27

55

5

55

142

58

62

58

58

5

59,62

59

77

161,162,163,166

44

155

144

11

11

12

12,13

10

75

44

22

198

AUTHOR 1 WVEX

Numbers indicate the pages of the text on which the author's work is

referred to.

Ahlberg, J .H.

Aitken 1 A.C.,

Boor, c. de,

Brocket, R.W.,

Bruijn, N.G. de,

cavaretta, A.S.

Cheney, E.W.,

Chui, K. 1

Coppel, W.A.

Cox, M.G.,

Davis, P.G.,

Feller, W.,

Fyfe, D.J.,

Golomb, M.,

Jakimovski, A. ,

Karlin, s., Koch, P.E. 1

Kolmogorov, A.,

Kreyszig, E.,

Kuga, K.,

Lancaster, P.,

Landau, E.,

Luenberger, O.G.,

Lyche, T.,

Marsden, M.J.,

Meinardus, G.,

Meir, A.,

Melkman, A. A.,

Michelli, C.A.,

55,81

83,84

6,11,12,17,45,46,156

135

48

103,120,152,154

98

103

9

38,40

10,25

31

55

144

81

6,11,17,19,21,33,152,153,154,161,162

40

102, 103

107

165

27

102

112

40,46,54

45

27,46

75

11

5,7,64,65,76,77,80,105,106,107

Morsche, H.G. ter,

Nilson, E.N.,

Nörlund1 N.E. 1

Northcott, D.G.,

Pinkus, A.,

P6lya, G.,

Rivlin, T.J.,

Roberts, A.W. ,

Russell, D.C.,

Schmidt, E.,

Schoenberg, I.J.,

Schönhage, A. I

Schumaker, L. 1

Schurer, F.,

Sharma1 A. 1

Smith, P.W.,

Ste~kin 1 S.B.,

Studden, W.J.,

Subbotinl Yu.N.,

Troch 1 I.,

Tzimbalario, J. 1

Varberg 1 D.E.,

Walsh1 J.L.,

Watkins, D.,

38144,46,52,62,65,67,75,76,1261130

55,81

58,139

133,144

22, 162

8

98, 105

178

81

27

7111112,17129,30,56,62,75177,10311191120

98

17

55172

75,1031119

103

1031105

17,19,1521161

81

9

103,119

178

55,81

27

199

200

SAMENVATTING

In hun eenvoudigste vorm zijn spline-functies {kortheidshalve splines)

functies die bestaan uit (stuksgewijze) polynomen, meestal van lage graad,

die in de zogenaamde knooppunten met een zekere mate van gladheid aanslui­

ten. Systematisch onderzoek naar het gedrag van splines dateert van 1946,

toen Schoenberg een aantal belangrijke artikelen over deze functies publi­

ceerde. Het blijkt dat splines, mede vanwege hun flexibiliteit, interessante

interpolatie- en extramale eigenschappen hebben, die hen ook uitermate

geschikt maken voor het numeriek benaderen van functies. Inmiddels is de

theorie en de toepassing van splines uitgegroeid tot een belangrijk (en

omvangrijk) onderdeelvan de approximatietheorie. Allerlei modificaties

hebben hun intrede gedaan. Zo zijn de polynomen vervangen door bijvoorbeeld

exponentiêle functies, of algemener door functies die in de nulruimte liggen

van een lineaire differentiaaloperator I.; de bijbehorende splines worden

daarom t-splines genoemd.

In dit proefschrift koment-splines aan de orde waarbij de differentiaal­

operator constante reêle coêfficiênten heeft, en dus geschreven kan worden

als pn(D) waarbij D de gewone eerste orde differentiaaloperator en pn het

karakteristieke polynoom van t is. Het eerste deel van het proefschrift

gaat over interpolatie eigenschappen van zogenaamde "cardinal" t. -splines,

d.w.z. splines waarvan de opeenvolgende knooppunten onderling dezelfde

afstand h hebben. Uitgaande van een reêelwaardige functie f op ~. een

lineaire differentiaaloperator t., en een vaste knooppuntafstand h, wordt

gevraagd een t.-spline met knooppunten O,±h,±2h, ••• aan te geven dief in

de punten a,a±h,a±2h, ••• interpoleert; hierbij is a een willekeurig gegeven

getal in (O,h]. De theorie van dit soort interpolatie voor polynomiale n splines, d.w.z. splines die behoren bij de operator pn(D) ~ D , is uitvoerig

behandeld door Schoenberg voor de gevallen a = ih en a = h. Een uitbreiding

van deze theorie tot willekeurige a € (O,h) is door Ter Morscha gegeven,

in eerste instantie voor periodieke cardinal polynomiale spline interpolatie.

In dit verband dient eveneens te worden gewezen op het werk van Michelli,

201

die cardinalt-spline interpolatie heeft onderzocht voor algemene operatoren

pn(D), waarbij pn alleen reële nulpunten heeft. In dit proefschrift wordt

cardinal t -spline interpolatie bestudeerd zonder deze eis voor pn. Een

belangrijk hulpmiddel daarbij zijn relaties tussen differenties en afgeleiden

van cardinal t -splines. Deze relaties laten zich op een overzichtelijke

manier weergeven met behulp van zogenaamde exponentiële cardinal !-splines;

dit zijn splines die, analoog aan de exponentiële functie, de eigenschap

hebben dat s (x+ h) Às (x), voor zekere À E :R en alle x E :R. De genoemde

relaties tussen differenties en afgeleiden worden ook toegepast bij het

onderzoek van periodieke !-spline interpolatie. Existentie- en eenduidig­

heidsstellingen en ook foutschattingen worden afgeleid, zowel voor cardinal

!-spline interpolatie als voor periodieke cardinal t -spline interpolatie.

Bij de foutschattingen gaat het om het bepalen van sup I f (x) - s (x) I over

alle (periodieke) functies f met llpn (D) fll ::;; 1. Hier en in het vervolg wordt

met II•IIJ steeds de supramumnorm op J c :R bedoeld; als J = R dan laten we + de index lR weg, als J m.0 := [0,00 ) dan noteren we 11•11+. Het blijkt dat

bij deze foutschattingen de zogenaamde perfecte Euler l-splines een belang­

rijke rol spelen. Dit zijn exponentiële cardinal !-splines E waarvoor geldt

f(x+h) =- E(x) voor alle x ER en lp (D)E(x) I = 1 voor alle x E :R behalve n

in de knooppunten.

Bij de uitbreiding van de theorie van cardinal !-spline interpolatie tot

operatoren pn(D), waarbij pn ook niet-reële nulpunten mag hebben, is de be­

langrijkste moeilijkheid dat een basis in de nulruimte van pn(D) niet op

elk interval een Chebyshevsysteem vormt. Op intervallen waar dit wel het

geval is kan, naar een klassiek resultaat van Pólya, de operator pn(D)

ontbonden worden in eerste orde differentiaaloperatoren van de vorm -1

Di = wi Dwi , waarbij wi (i = 1,2, ••• ,n) op het betreffende interval

positieve functies zijn; er geldt dan pn(D) Dn Dn-l ••• D1• Een dergelijke

ontbinding is van belang voor het tellen van nulpunten van !-splines; de

stelling van Rolle en de stelling van Budan-Fourier als verfijning daarvan

kunnen eveneens geformuleerd worden voor o1,D2 , ••• ,Dn.

Het tweede deel van het proefschrift gaat over extramale eigenschappen van

perfecte t-splines met betrekking tot zogenaamde Landauproblemen. Ih 1913

bewees Landau voor tweemaal differentieerbare functies f op :R de ongelijk-+

heid lif"ll s 12 filflllif" 11; voor functies f gedefinieerd op de halfrechte :R0 bewees hij dat geldt llf'll s 2fllfll llf"ll. Deze ongelijkheden kunnen niet

+ + + ' verscherpt worden, d.w.z. de constanten 12 en 2 kunnen niet worden vervangen

202

door kleinere getallen. Landau's eerste ongelijkheid werd in 1939 door

Kolmogorov gegeneraliseerd tot hogere afgeleiden. Op J:<~ is een dergeli·jke

generalisatie van Landau's tweede ongelijkheid ~eel wat gecompliceerder.

Uiteindelijk is dit prÓbleem opgelost door Schoenberg en Cavaretta in die

zin, dat zij zogenaamde extremale funaties bepaald hebben; dit zijn functies

waarbij de betreffende ongelijkheden in ge'lijkheden overgaan. Bovenstaande

problemen zijn bijzondere gevallen van het algemenere Landauprobleem dat we

nu beschrijven.

Laat J een willekeurig gesloten deelinterval van J:< zijn, en pn(D) en pk(D)

lineaire differentiaaloperatoren van de orde n enk~ n-1; bepaal dan bij

gegeven m > 0 het supremum van llpk (D) fll J over alle functies f die (n- 1)

keer differentieerbaar zijn op J, een absoluut continue (n- 1) -ste afgeleide

hebben en waarvoor geldt dat llfiiJ ~men llpn(D)fiiJ s 1. Deze klasse functies

geven we aan met Fm (pn,J). Een functie f waarvoor het supremum van I!Pk (D) fiiJ

wordt gerealiseerd wordt weer een extremale functie genoemd. Aldus geformu-+ leerd blijkt in de gevallen J =Ren J =m0 het Landauprobleem equivalent

te zijn met het probleem het maximum te bepalen van een lineaire functie (n-1) T over de verzameling van alle punten (f(O),f'(O), ••• ,f (0)) ,

met f €: F (p ,J}. We besteden daarom in het bijzonder aandacht aan de m n +

gevallen J =Ren J =1:<0 ; de zojuist genoemde verzamelingen worden dan

respectievelijk met r (p ) en r+(p )) aangegeven. Een aantal ~peciale m n m n gevallen van het Landau probleem wordt opgelost door middel van een

parametrisering van de verzamelingen r (p) en r+(p ). m n m n

Wij besluiten met een korte opsomming van de inhoud van de verschillende

hoofdstukken.

Hoofdstuk 1 heeft een inleidend karakter. Het bevat de definities van de

verschillende soorten splines die in het proefschrift een rol spelen.

Verder komen o.a. aan de orde: gedisconjugeerdheid van een differentiaal­

operator, verschillende soorten Chebyshevsystemen en gedeelde differenties.

Dit alles is standaardmateriaal in de approximatietheorie. Wel nieuw is een

generalisatie van de stelling van Budan-Fourier voor functies met stuksge­

wijze continue afgeleiden.

In Hoofdstuk 2 worden B-splines behandeld,behorende bij willekeurige diffe­

rentiaaloperatoren pn(D). Eerst besteden we aandacht aan een aantal belang­

rijke eigenschappen van deze nuttige functies. Vervolgens wordt met behulp

van de stelling van Budan-Fourier de zogenaamde "totale positiviteit" van

een rij van opeenvolgende B-splines bewezen. Via de Fourier-:gf;ltransforme.erden

203

van B-splines worden eenvoudige recurrente betrekkingen voor B-splines

afgeleid. Een daarvan wordt gebruikt om de verdeling van knooppunten te

bepalen waarvoor de supremumnorm van een polyn.omiale B-spline minimaal is.

Hoofdstuk 3 bevat de reeds eerder genoemde relaties tussen differenties en

afgeleiden van cardinal t-splines, die in het eerste deel van het proef­

schrift fundamenteel zijn bij het onderzoek van (periodieke) cardinal t­

spline interpolatie. Voor een eenvoudige representatie van deze relaties

worden exponentiële t-splines gebruikt. Aan de eigenschappen van deze

splines wordt ruime aandacht geschonken.

Existentie- en eenduidigheidsstellingen bij (periodieke) cardinal t-spline

interpolatie zijn een onderdeel van Hoofdstuk 4. Tevens worden daar (opti­

male) foutschattingen afgeleid. Hoewel bij periodieke cardinal t-spline

interpolatie een beroep kan worden gedaan op resultaten voor cardinal t­

spline interpolatie, ontstaan er met name bij optimale foutschattingen

analytische problemen. Hieraan wordt ook aandacht besteed.

In het begin van Hoofdstuk 5 wordt de betekenis van het Landauprobleem

aangegeven voor de bepaling van optimale differentiatie-algoritmen. Verder

worden algemene eigenschappen van de verzamelingen rm(pn) en r:(pnl afgeleid.

In het bijzonder wordt aandacht gegeven aan de gevallen n 2 en n = 3. We

tonen aan dat bij de parametrisering van de rand van rm(pn) en r:(pn)

perfecte Euler t-splines een fundamentele rol spelen.

Het Landauprobleem voor periodieke functies is het onderwerp van Hoofdstuk6.

Hier wordt o.a. een verregaande generalisatie gegeven van een resultaat van

Northcott, dat betrekking heeft op het maximaliseren van 11 f (k)ll ( 1 ~ k ~ n-1)

voor periodieke functies f met 11 f (n)ll ~ 1. + In Hoofdstuk 7, tenslotte, wordt het Landauprobleem op de halfrechte E 0

beschouwd voor algemene pn(D), waarbij echter t.a.v. het polynoom Pn wordt

verondersteld dat het uitsluitend reële nulpunten heeft en dat bovendien

geldt dat pn(O) = 0. Met behulp van een representatiestelling voor een

geschikte klasse van perfecte t-splines tonen we aan dat een extremale

functie wordt verkregen als de limiet van een rij van geschikte perfecte

t-splines, die een specifiek oscillatiegedrag vertonen. Op grond van de

bereikte resultaten kan de rand van de verzameling r+(D 3) in detail worden m + 4

beschreven. Bij een parametrisering van de rand van de verzameling r 1 (Dl

doen zich echter reeds aanzienlijke analytische moeilijkheden voor. Een

punt van deze rand wordt numeriek bepaald.

204

CURRICULUM VITAE

De schrijver van dit proefschrift werd op 31 december 1944 geboren te

Enter. Vanaf 1957 bezocht hij de Pius X-HBS te Almelo, waar hij in 1962

het diploma HBS-b verwierf. Daarna studeerde hij Wis- en Natuurkunde aan

de Katholieke Universiteit Nijmegen. In december 1967 werd het doctoraal

examen Wiskunde behaald. Het afstudeerwerk, dat betrekking had op een

onderwerp uit het gebied van de niet-lineaire differentiaalvergelijkingen,

werd verricht aan de Technische Hogeschool Eindhoven onder leiding van

Prof.Dr. N.G. de Bruijn en Prof.Dr.Ir. M.L.J. Hautus (toen Ir. Hautus).

Sinds 1968 is hij verbonden aan de Onderafdeling der Wiskunde en Informatica

van de Technische Hogeschool Eindhoven, waar hij als wetenschappelijk

hoofdmedewerker deel uitmaakt van de Groep Basisonderwijs. Tevens is hij

lid van de vakgroep Toegepaste Analyse, Mathematische Fysica, Mechanica

en Numerieke Wiskunde, waar hij zijn werkzaamheden verricht in een overeen­

stemmingsrelatie met Prof.Dr.Ir. F. Schurer.

STELLINGEN

bij het proefschrift

INTERPOLATIONAL ANI> EXTREMA!. PROPERTIES

OF J: -SPLINE FUNCTIONS

van

H.G. ter Morsche

1

De Euler !-spline '!' 1

(p ;h,•} en de perfecte Euler !-spline f(p ;h, •) hebben - n n

respectievelijk de eigenschappen

-n+l lim h '1'_1 (pn;h,ht) ~

n '1'_

1 (D ;1,t} ,

( t <- lR} •

lim h-n f(p ;h,ht) = E(Dn;l,t) • h+O n

Subsecties 3.2.3 en 3.2.7 van dit proefschrift.

2

(n-1) · Zij f E C (lR) met absoluut continue (n- 1) -ste afgeleide. De interpole-

rende cardinal !-spline in Theerem 4.4.2 van dit proefschrift heeft de

eigenschap dat voor alle x E lR

(h "' 0} •

Sectie 4.4 van dit proefschrift.

3

Zij pn een monisch polynoom van de graad n met uitsluitend reële nulpunten

en zij E(p ;h,•) de bijbehorende perfecte Euler !-spline. Dan is n

suptEIR IE(pn;h,t) I een continue en monotoon stijgende functie vanhop

[O,~); dit kan eenvoudig worden aangetoond met behulp van een resultaat van

Sharma en Tzimbalario.

Subsecties 3.2.7 en 5.5.2 van dit proefschrift.

Sharma, A.; Tzimbalario, J.,Landau-type inequalities forsome linear differential operators. Illinois J. Math. 20 (1976}, 443-455.

4

Zij!: ~+~n (n ~ 2) een differentieerbare functie met absoluut continue

afgeleide Zij verder I f (t) I := Vf21 (t) + ••• + f 2 (t) 1 waarbij f 11 ••• 1 f - n n de componenten van ! zijn.

Als lfCtll::; 1 en l!"(tll::; 1 voor (bijna) alle t € ~.dan geldt

I!' (t) '2

::; 1 + Yl- l!<tll2

(t E ~}

en deze ongelijkheid kan niet verscherpt worden.

Schoenberg 1 I.J., The Landau problem for motionsin a ring and bounded continua. Amer. Math. Monthly 84 (1977), 1-12.

5

Ge • • 2 + 4 ul t 11 ll ll lfl UI geven z~Jn n ree e ge a en Yoi••·•Yn•Yo•···•Yn•Yo •Yn •

De polynomiale vijfde-graads spline s E c3t[0,1]) met knooppunten van

C''illiplici tei t twee in .!. , ~ 1 ••• 1 n- 1 die voldoet aan de interpolatievoor-n n n ;qaarden

s (.!) = y n \1

(\I 0,1, ... ,n),

s"' (0) y'" 0 s"' (1) = y~"

kan op eenvoudige wijze worden geconstrueerd door gebruik te maken van

voortbrengende functies.

Morsche, H.G. ter, On the construction of a {0 1 2)-interpolating deficient quintic spline function. TH-Report 74-WSK~Os, Eindhoven Univarsity of T~ch~ol0gy, Eindhoven, 1974. --

6

Zij M de klasse van functies f gedefinieerd op (0 1®) en met de eigenschappen

i) fis integreerbaar op ieder interval [a,b] c (O,~>~

ii) er bestaan positieve constanten a en B zodanig dat

f(t) = 0(t-a) (t+O) I f{t) = Oce 6t) (t +co) •

Zij verder {An)7 de rij van positieve lineaire operatoren gedefinieerd door

f n-1 -nt t e f(xt)dt (x > 0) •

' 0

De saturatieklasse van (An)~ met betrekking tot M bestaat dan uit functies f

waarvan de afgeleide locaal Lipschitz-continu is, en de saturatieorde be­

draagt 1/n, d.w.z. op ieder compact interval van (0,~) is

(x) - f (x) I = 0 (.!_) n

(n + oo)

dan en slechts dan als f tot de genoemde deelklasse van M behoort.

Morsche, H.G. ter, The saturation class of the Gamma operators. Memorandum 78-06, Eindhoven University of Technology, Eindhoven, 1978.

7

Laat Y1,Y2 , ..• onafhankelijke en identiek verdeelde stochastische variabelen

zijn met een verdelingsfunctie Fx' die afhangt van een parameter x E [a,b].

Zij !b dF (T) = 1, zij E de verwachtingsoperator met betrekking tot Fx en a x x

zij Ex Yn x (x <' Ca,b]; n E :N). Als de rij (Ln) 7 van positieve lineaire

operatoren gedefinieerd is door

met X n

(Y1 +Y2 + ... +Yn)/n en f E C([a,b]), dan is de saturatieklasse 00

van (Ln)l de verzameling functies op [a,b] met een Lipschitz-continue 1 2

afgeleide, en de saturatieorde is n E (Yn - x) •

Morsche, H.G. ter, Saturation in c[a,b] of a special sequence of linear positive operators. Memorandum 78-05, Eindhoven University of Technology, Eindhoven, 1978.

8

Zij m e: JN en zij (an) :00

een rij complexe getallen met de eigenschap dat

voor alle n E 1Z.

2 ~ 2m , 1 •

Dan geldt

(n e: ?Z) ,

en deze ongelijkheid kan niet verscherpt worden.

9

De door Sierpinski gegeven oplossing van het probleem:

3 3 3 bewijs dat voor m = 1 en m == 2 de vergelijking x +y + z = mxyz geen oplos-3 3 3 singen x,y,z in :!f heeft, en bepaal alle x,y,z in :!f waarvoor x +y +z

= 3xyz,

kan aanzienlijk vereenvoudigd worden als men gebruik maakt van de identiteit

3 3 3 x +y + z - 3xyz 2 2 2 ! (x + y + z) ((x- y) + (y - z) + (z -x) ) •

Sierpinski, w., 250 problems in elementary number theory (problem 156). Amer. Elsevier Publ. Co., New York, 1970.

10

In onderwijskundige verhandelingen over hoorcolleges wordt te weinig aan­

dacht geschonken aan de medeverantwoordelijkheid van studenten voor de

kwaliteit van deze onderwijsvorm.

Hout, J.F.M.J. van; Mirande, M.J.A.; Smuling, E.B., Geven van hoorcolleges. Aula 807. Het Spectrum B.V., Utrecht, 1981.

11

Een aantal belangrijke beleidsvoornemens met betrekking tot de financiering

en de organisatie van het universitaire onderzoek in Nederland, zoals

geformuleerd in de beleidsnota universitair onderzoek, staat op gespannen

voet met het in dezelfde nota geformuleerde uitgangspunt, dat de universi­

teiten en hogescholen een zelfstandiger beleid moeten kunnen voeren.

Beleidsnota universitair onderzoek. Tweede Kamer, zitting 1979-1980, 15825, no. 1-2.

12

Het opzettelijk en bij herhaling veroorzaken van een buitenspelsituatie bij

het voetbalspel is een vorm van spelbederf en Z()U als zodanig door de

scheidsrechter bestraft dienen te worden.

H.G. ter Morsche

16 april 1982