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Transcript of Non-linear difference equation
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O n
nonlinear
d i f ference
equations
By
L. J.
G R I M M *
and W. A.
H A R R I S , Jr.**
Introduction.
We consider a system of nonlinear dif ference equations of the
form
(0.1) y(x +
l~)=f(x,y(x-)-),
x\
where x is a complex variable,y an ^-dimensional vector, and / an
n-dimensional vector.
/^
Each component
of the
^-dimensional vector
/ is
assumed
to be
holomorphic in a region
R =
S
Q
X C7
0
,where
fo r
some positive constants a, d
Qy
p
Qy
where the norm of a vector u is
n
given by W =S|w,-|.
Let
(0.2)
be
the
expansion
of / in
powers
ofy
ly
-',y
n
,
where
p is a set of
non-
negative
integersp
lf
--, p
n
, B(#) an nXn
matrix,
/
0
and /p
^-dimensional
vectors, and
(0.3)
Received November 17 , 1965.
*
Department
of Mathematics,
University
of Utah, U. S. A.
**
School
of
Mathematics, University
of Minnesota, U. S. A. and
Research Insti-
tute
for
Mathematical Sciences,
Kyoto
Universi ty,
Japan. U. S.-Japan
Cooperative
Science Program,N SFGrant
GF-214.
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We
shall suppose that f
0
, fy, B
are
holomorphic
in
S
Q
and
have
the
21 2 L.
/. Grimm
and W. A. Harris, Jr.
We
shall suppose that
asymptotic expansions
(0.4)
as
x
approaches
infinity
through
the
sector
5
0
.
In order to construct solutions of (0. 1) it is important to
obtain their first approxim ation ; in general
this
is
difficult.
However,
if
a solution y(x) has a limit
y
Q
as
x
approaches infinity and
/(, 3 > o ) is
defined, then y
Q
must satisfy
the
equation
J^/C
00
,^)-
O n
the other hand, if there exists a
y
Q
which satisfies thisequation,
y
Q
=/(,yo), then, und er suitable stability con ditions, it may b e
expected that every solution
in a
neighborhood
of
y
Q
will approach
y
Q
as x approaches
infinity.
The purpose of this paper is to show
how, un der suitable hypotheses, to con struct the gen eral solution of
the system (0 . 1) in a region of the
form
(0.5)
/
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On nonlinear difference equations
213
(0. 4) as x approaches infinity
through
the sector Si. Suppose
x\
that B
Q
and B
0
I are
nonsingular.
Further
assume
/ o o = 0 , and
also
that if A
i9
i =l,---,n, are the
eigenvalues
of B
Q
,
then
0^arg
(
log^-)- Then
if the
positive constants a >
2
and p^
1
are
sufficiently
large,
there
exists a solution
(0.7)
y=*W
of (0. 1)
such that < ( # )is analytic and admits the asymptotic
expansion
(0.8) ^OO^SAJr"
i / = i
as
x
tends
to
infinity
in the
domain,
S
2
:
|
arg
(#^
-tf
2
)l
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21 4
L. /. Grimm and W. A.
Harris,
Jr.
S
3
: arg(*0-
f
'-0
8
) 0, p
3
>0,
Z i
and let A
possess
the
asymptotic
expanion
as x approaches infinity through the sector
S
3
.
Suppose
that
vi,
-,&
are the
distinct eigenvalues
of A
0y
that none
of
these
is
zero,
and
that
0 = a r g (
log
(/,-//*/)),
i j.
Suppose
further
that
A
Q
has the
block-diagonal (Jordan) form A
Q
=
diag(Ai,>-,A?), where
A^jujlj + Nj, with Ij the m
r
dimensional identity matrix and Nj a
nilpotent matrix.
Then thereexistsamatrix TOOwithcomponents
holomorphic
in some sector,
S
4
: largC^-^-flO|
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On
nonlinear difference
equations
215
that
th e
matr ix A(x")
has the
block-diagonal
form
of
J3(#).
Let
(0.17) z = P(x,u')
be a transformation of the vector
z
such that
P(x,u)
can be repre-
sented
by a
uniformly convergent
series of the
form
(0.18)
in a
region J?
4
= S
4
X
C 7
4
given
by
U
t
:\\u\\0,p
4
>0,
with
5
4
>0 sufficiently
small, with
coefficients
/\
holomorphic
for
x^S
4
,
an d
admitting asymptotic expansions
x
as
^
approaches infinity through
5
4
.
W e are now in a
position
to
prove
our
main theorem.
Theorem
3. Suppose that the
matrix
A
Q
has eigenvalues
satisfying
(0.19) 0
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21 6 L. J. Grimm and W. A. Harris, Jr.
where 2
5
>0 and 0
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On nonlinear difference equations 21 7
In particular,
if the
/l/s
are all
distinct,
the
system
(0. 21)
becomes scalar equations and the system can be solved recursively.
If ,
in
addition, ^yp^l
for all
indices j
and p, the
system (0.22)
has
diagonal homogeneous
fo rm
(0.25) w,(* + )= 0u00 w,00-
Af te r
obtaining the general solution of the system (0. 22) , we
can construct
the
general solution
of (0. 10) by
substituting
the
solution
of (0.22)
into
the
transformation
(0.17).
In
doing
so it is
necessary
to
estimate
the
magnitude
of the
solution
of the
system
(0.22).
If the
reduced system
(0. 22) is
normal
in an
extended
sense (we
shall give a precise def in i t ion of this concept in Section 7) we shall
find a region of the type
in
which
the
solution
is
uniformly
bounded
and
approaches zero
as x
tends
to
in f in i ty
in this
region. Choosing 0consistent with Theorems
1-3, the general solution of the original system (0. 1) is given in
this
region by
(0.26) X*)=000+P(*, t/(*,C(*)),
where U(x,C(#))
is the
general solution
of the
reduced equation
(0. 22) and COO is an arbitrary bounded periodic vector with period
one.
Thus
we have attained our main objective.
The
scalar case, n = l,
has
been treated
by J.
Horn
[9]
under
x
the assumption that f(jx, y) is holomorphic for I%I^>R
0
, \ y \
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218 L. J. Grimm and W. A.
Harris,
Jr.
an
upper half-plane, whi le
w e
have established
th e
convergence
of
this
series. General results
of
essentially
th e
same nature
as
ours
have been obtained by W. A. Harris , Jr. and Y. Sibuya [7] in half-
planes of the for m ] Im x \ > a u n de r the condi t ions
U i | = l , and h U , | * ' = M , |
j
= l
fo r
all
fa.
O urresults for
non l ineardifference equations parallel similar
results
in the
theory
of
ord inary
differential
equations
for
systems
of
th e form
fo r
which
th e
corresponding linear system
--*.
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On nonlinear difference equations
219
(1.2) X*)=*(*)+*(*)-
Then
X* + )=/(#, X*))
becomes
(1.3) *(* + ) + *(* + )
=/(*,* (*)+(*))
and,
since 000 is a solution, (1. 3) becomes
(*)[*(*)+0(*)]P
which can be written as
(1 .
4) *(#4-i)
=400*00 +/(*, *00) =400*00
where
th e series is convergent for x0, d
z
sufficiently
small, and where the coefficients A(x) and
/p(^)
are
holomorphic for x^S
z>
and have asymptotic expansions
(1. 5)
as x
tends
to in f in i ty
through
the
sector
S
2
.
Equate
the
linear terms
in
(1.3)and
(1.4),
and
using
the
fac t that 0(^)=0(^"
1
)
we
obtain
(1.6) A =
fio(=/,(~,0)).
b) Remarks on Theorem 1. The region of validity of solutions
obtained
in
Theorem
1 is the
sector
S
2
of the
fo rm
(1.7)
where
0 an d p may be
described
as follo w s: In the
complex
d raw all the points
l o g U i l iarg^-,
i =l,'~,n. Then draw rays through each
of
these points extending
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220
L. J. Grimm and W. A. Harris, Jr.
from
th e
origin
to
infinity.
These
rays will divide
th e
plane into
a
countable number
of
sectors,
Si, Sa, The
conculsion
of
Theorem
1 holds for all choices of 0 and p , p>0 , such that the sector
is contained in some sector Sy-
Exactly
one of the
sectors
S/,
call
it
X
0
, will have
one of the
following
properties:
i ) The
positive real axis will
be
interior
to S-
ii) The positive real
axis
will be the lower boundary of S
0
,
i. e., S
will
be a
sector
of the
form 0 < a r g C < C o
fo r
some Co>0.
It
is
clear that case
ii )
will hold
if, and
only
if, at
least
one of the
eigenvalues h satisfies 0
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On nonlinear difference equations
221
Case ii) 6 ^ 0
B
Figure
1.
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22 2
L.J. Grimm
and
W.A, Harris, Jr.
inf in i ty.
However, the restrictions in Theorem 3, |0|+p
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On nonlinear difference equations
223
Let
A, B, Q
have
the
parti t ioning
A
=
induced by the parti t ioning of A
0
. If there is a solution of the
desired
form
then ,
'^11 A
X\ x\
:
A
n
A
r
B=
Bl1
.
0
0 D
n
rr
j
, G
=
L ?
1 2
'
Q l r
, G
r t
b ,
and the equat ion for determining
Q
becomes
(2.
7)
AQuA,=AMu-Qu
If Q is determined in
this
m a n n e r , th e n 5 and T are also determined.
Equ ation (2. 7) is a system of no nl in ear
difference
equations of the
form
(2.8) ^(^)=^(^,^^)=^o*W+C * ( ^ ) 3;+A*(^^^)
where
the
components
of the
vector h*(x,y,Ay)
are
polynomials
in
y an d
Ay
w i th
coefficients
that are
O C ^ r
1
) -
Hence, for
\x \ sufficiently
large,
#eS
3
, i.e.,
in some sector
fo r
a
e
>a
3
, we may rewri te th e system (2. 8) in the
form
(2 .9 )
4y
where
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224 L.
/.
Grimm and W, A. Harris, Jr.
are & /
&,
i,j =l,--,r. Thus the problem of block-diagonalization
has been reduced to that of finding a solution of a system of nonl inear
difference equations
of a
form
to
which
Theorem 1 is
applicable.
Applying Theorem 1 w e obtain a solution Q(#) of equation (2. 7)
/x
in a sector S
4
cS
6
,
arg(xe~*
- a*)\
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O n
nonlinear difference equations 225
500=2.8,*-'
= 0
(3.2)
as x approaches infinity
through
the sector S
B
. Suppose further
that BQ is
nonsingular
and that the eigenvalues of Bo
l
A
0
have
absolute value less than 1. Then there exists a unique bounded
holomorphic solution
y of
(3. 1)
in
some sector
S
Q
: \
arg(xe~
ie
-
0
8
) |.a$,
and possessing
there
the
asymptotic expansion
as
x
approaches infinity through
the
sector S
6
. Further,
there
exists a
constant
C depending only upo n the matrices B(x) and
A(x) ,
such
that for
(3.3) b
Proof:
Since
B
Q
is nonsingular , for
% ^S
5
, \x\ sufficiently
large,
x)
wi l l
exist.
Wri te
(3. 1) as
(3.4)
Since by hypothesis the eigenvalues of B^Ao have mo dulus less than
1, there exists
a
nons ingular cons tant matr ix P such that
(3.5) \ \P-*B^A*P\\
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226
L. J. Grimm and W. A. Harris, Jr.
Define
an d let y = Pz.
Then (3.4) becomes
(3 .7 ) z(x)=R(
Let
L
=
sup||/00||,
S6
an d
M K L
V L
1
r
Let $ be the
fami ly
of all
m-dim ens ional vec tor fun c t ions
q >
(x) holo-
morphic forx S
6
, such
that
| |0>(#) | |
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On
nonlinear 'difference equations
227
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228 L. J. Grimm and W. A.
Harris,
Jr.
of
the magni tude of the r>. Write
Then
(4.3) || S
rqM || , /= , - , # , | t > [ =&. We
apply Theorem
1 as
above
t o
obtain
P(k, #) and G(k , x ) for kN
0
in a
sector
fo r some constants 0
9
( A ) an d
I t is im por tan t to show th at there is a single region of this form
in
which all the P(k, #) exist.
This
will be the case if we can show
that
\\A~\k,
x) 'C(k,x) ||
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On nonlinear
difference
equations 235
in the uniform region S
9
(Af
2
) which we write as
Further, there exists
a
constant depending only
on A(x)
(with
0
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236 L. J.
Grimm
and W. A. Harris, Jr.
to
yield
(6.
3)
1 0 1 = 2
where
g(x,y)
and
h(x,y)
are analytic for #e
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On nonlinear difference equations 23 7
(6.7)
l
Suppress
the
a rg u ment
of w and
rear range
to
obtain
(6 .8) S Q
p
(^+l)(^(^)u;)P-^U)(S QpW^)
I P I ^ ^ V |p | ^JV
,R(x,
After
substi tut ing
the
representat ion
for
R(x,w)
given
in
( 6 . 4 ) ,
w e
obtain th e following formal
representat ions:
W e notice in part icular that th e system of equations for the Qp
obtained
by
equating coeffic ients
of w% is
precisely
the
same
as the
system
for the Pp
except that
th e
nonhom ogenous te rm
is different.
Since [ p |
>N>N
2
,
th e
p's
can be determined in a un i form region
(6.9) S
2
:|arg(*^"-\
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238 L. I Grimmand W. A. Harris, Jr.
(6.10) S3HCP(*)|I
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On nonlinear difference equations 239
an d
from
R(x
y
w^=w-\-
2 Q$(x}w$,
defined
the
func t ions
(6.
12)
R?(X,W)=WJ
+
S |QX*)#, and
R(x,W =W
1
+'-'+W
n
Define (^, v)=1?(^,5) with # as above.
Then
Then
th e
vector
(6.
13)
has
i'ih component
Then
(6 . 14)
By definition
i. e., the
coefficients
of z
k
in the
multiple power series
fo r gf are
positive
and not
less than
the
absolute values
of the
corresponding
coefficients
of the
series
for
gj. Similarly,
for all
j,
*(x, z)
for ail ;,
Thus
m,(x, w)= g> (x, R(x,
where R(x,w)
= (R,
,
i? )
r
. Since
all
terms
of
gf
are
positive,
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24 0
L. j.
Grimm
and W. A.
Harris,
Jr.
Sum
over
j to
obtain
(6.15) m(x,0))=(*,
Let Wi
= v, i =
l,---,n.
Then (6.15) becomes
K#,
*0
as s
A(#,
C 6 = N
Notice that,
because
of the form ofR(x,w}, Jh
a
(x)=
| P =
for oj= 2, ,
J V 1 ,
and that m
N
(x)=0. Hence S maWv is a
Q 5 =
JV
+ 1
majorant fo r m(x,
v} g(x, v), since g(x
9
v)
is a
polynomial
of
degree
at most
N 1
in
v,
and the terms in
m
of degree less than
N+1
ar e
independent of the Q p,and are
hence equal
to the
corresponding^'s.
Recall that
R x,M ; ) =
an d
that
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O n nonlinear
difference
equations
241
Since ||.4( )ll
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242
L. J.
Grimm
and W. A. Harris, Jr.
W e
shall show
that
(6. 19) has a uniqu e fo rmal solution of the form
(6.20)
f
Then define g
k
an d
h
k
by
(6.21)
Substitute
(6. 20)
into
(6. 19) to
obtain
the
fo rmal equation
Since
this is to be a
fo rmal ident i ty
in
v, equate coefficients
of
v
k
to obtain
(6.
22)
f
4
*
=C
[
G
k
n
k
+
n
k
R
k
(O -
G
k
n
k
+
n
k
S
k
(f
a
) +
f
VA +n
k
T
k
(f
a
) ] ,
where
j ? ? * ,
S*, and T
fe
are d efined by
S n*
T
t
v"
=
=AT * =
JV
w = 2 fe =
JV
Notice that R
k
, S
k
, and T
fe
are all polyn om ials in the
a
(
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O n
nonlinear
difference equations. 243
W e now
show that
(6. 20 ) is a
ma jo ran t
for
Q
y
i. e.,
that
The
proof proceeds
by
induction. First, notice that
for
k = N,
R
N
=
0
S
S
N
= h
N
, T
N
= 0. Since 0^00=0,
n
N
R
N
^m
N
.
Also
n
N
h
N
^
00^ ^
I N ( X ^ ) ,
since ^H
k
t
k
^>h(
9
), and the term h
N
is independent of the
k
= N
~ X\
f's, while /jv is independent of
p.
Since the Q's were determined in
( 5
2 j
,
and
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244 L. J. Grimmand W. A. Harris, Jr.
by
subtracting the terms in the components of Q
a
, \a\=m
f rom
^ ). Recall that
N
1 N
If weexpand SftG^w+ S G
a
v
a
y in powers of #, weobtain
k
= N 05
= 2 =A T
where #
A
is a polynomial in f
a
for a
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O n nonlinear
difference
equations 245
However,
since by the induction hypothesis,
S I I Qp(# +1)11
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246
L.
/. Grimm
and W. A.
Harris,
Jr.
in an
essentially similar fashion
to
yield
%
m
J?
w
I>w
m
(:r )=
Thus
th e
estimate (6. 10 ) yields for all
S \\QvW \\^C(n
n
R
n
+n
m
S
m
+n
n
T
n
)
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On nonlinear
difference
equations 247
CO=
Clearly the mapping is welldefined if 8
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248 L. /.
Grimm
and W. A. Harris, Jr.
with
Ni th e
n i lpotent matr ix defined
in
( 2 . 2 ) .
The
nex t
k
2
k i
systems, corresponding
to the
indices
j =ki+l,ki + 2,,k
z
, arenon-
homogeneous systems
of the
form
(7. 2)
where
the
components
of g
j
ar e
polynomials
in the
components
of
u
l
,---,u
kl
.
Hence
the
general solution
of
(7.2)
can be
obtained
by
obtaining
the
general solutions
of all of the
systems
(7. 1) and
utilizing these
to
evaluate
the
func t ions
g
j
. Let g
j
(x} be the g
j
evaluted
in this way. The
remaining systems
forj = k
z
+ l, - are of
form
analogous
to
that
of (7. 2) , and we
proceed
in the
m a n n e r
described above to find the general solution of the redu ced system
(0.22).
Thus
the problem of solving (0 . 22) falls n atura lly into tw o parts,
the solving of l inear homogeneous equations and of l inear
nonhomo-
geneous of the forms
(7.3) (# + ) =4(*)(#), and
(7.
4) (* + )
=4(*)M(*)+(*),
respectively, where
A(#)=jul+N+f]A'x-,
s=l
where N
has the
form
of
N
{
above.
We consider the homogenous case (7. 3) first : A system of the
form (7 . 3) is called normal if there exist a formal
fund amen ta l
mat r ix
of the
form
where
R
is a cons tant m atr ix . O therwise the system (7. 3) is called
anormal.
If all of the
corresponding l inear homogeneous systems
are
normal ,
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On nonlinear difference equations 249
we
m ay
assume that
all of the
nilpotent matrices N
are
zero, since
Harris [4] has shown that this may be effected by a linear trans-
formation
which
is a
polynomial
in X'
1
with determinant
n ot
identi-
cally
zero. Further,
it is
known [1] ,
[5 ]
that
in the
normal case
there
exist
analytic
fundamenta l
matrices which have the
formal
fundamenta l
matrices as asymptotic representations in right
half-
planes. Hence the behavior of the fu n da m en tal m atrix as x tends to
infinity is essentially determ ine d by //, but since 0
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250
L.
/. Grimm
and W. A. Harris, Jr.
are
normal;
ii) if j u 7 * #
for
some r/ and integer 0/ for all j\
iii) there
exists
a fo rmal particular solution of the form
(log )
7
^00> where
Hence by the
results
of
Harris
an d
Sibuya
[5 ]
there
exists an
analytic
solution asymptotic to this formal solution in a sector of the form
(7. 5)
o This
particular solution has the same rate of growth as the
solution
of the
corresponding homogeneous equation.
By
induction
the
general solution
of the
reduced equation
(0. 22) can be
thus con-
structed in a region of the form (7 . 5) , if the reduced equation is
normal
in the extended sense, an d will have properties similar to
those
of the
general solutions when
th e
reduced equation
w as
linear.
W e
m ay
summarize
our
results
in the
following:
Theorem
4
0
Let the reduced equation
(0. 22)
beeither linear
or normal in the extended sense. Then the general solution of
(0.22) can bewritten in the form
(7 .6)
M 00 =*/(*, COO)
in a sector R of the form
where U(x, COO) is holomorphic in R, tends uniformly to
zero
as
x
approaches infinity through
R, and
possesses
an asymptotic
expansion
in
this region,
and
COO
is an
arbitrary bounded
periodic
vector of period 1 .
Hence, if 0 is chosen sufficiently small 0;>0, an d compatible with
the
hypotheses
of
Theorem
1
9
2, and 3,
(using,
for
instance,
the
sector
S defined in
Section
1 to
choose
0)we can
combine Theorems
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On
nonlinear difference equations 251
1 , 2
9
3
9
and 4 to
obtain
in
this case
the
general solution
of (0. 1) in
th e form
X* ) = 0 0 0 + /> ( * , OX*, C O O ) ) .
8.
General remarks-
If the eigenvalues /I/ of the m atrix A
0
satisfy K |^ | , s imilar
results corresponding to Theorem 3 are available in sectors which
cover a region of the
form
0 < < a r g ( ^ 4 - ^ ) < 2 7 r ,
a>0 .
If we assume the existence of a particular solution, or
/(#,0)= -
0 ,
an d
then
by
choosing either
Ui=--=u
p
=
Q, or u
k
+i=
m
-=u
n
= Q, similar
results are available where now C OO will be either an n p or k
dimensional arbitrary periodic vector .
The possibility of obtaining the uni form asymptotic expansion
fo r the t ransformat ion P(x, u) has been demonstrated by Harr is and
Sibuya [7 ] und er more restrictive hypothesis including
the
uniform
asymptotic expansion
We shall treat this question in a subsequent paper.
O ne
would
expect that
the
results embodied
in
Theorm
4 are
valid wi thout
th e
restriction
:
n o r m a l
in the
extended sense.
BIBLIOGRAPHY
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Bi rkho ff ,
General
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Arner.
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(1911).
[ 2 ] G. D. Bi rkho ff and W. J. T r j i t z i ns ky , Analytic
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