Spectral Envelopes of Integer Subsets

2nd Asian Conference on

Nonlinear Analysis and Optimization

Patong beach, Phuket, THAILAND

September 9-12, 2010

Wayne Lawton

Department of Mathematics

National University of Singaporematwml@nus.edu.sg

http://www.math.nus.edu.sg/~matwml

Characters

ZnTTen ,:

CRZ integer, real,

ZRT / circle group (real)

Tttkin ete ,)( 2

characters (complexexponential functions)

complex numbers

1||: wCwT circle group (complex)

Polynomials

Fk j

jmk

k wcwwcwL )()( CCL }0{\:

ZF finite subset

CTf :

Laurent polynomial

Fk

iktkecteLtf 2

1 )()(

trigonometric polynomialwhose frequencies are in F

Theorem 1 (Jensen)

1

01||

|||)(|logexpj

jcdttf

Spectral Envelopes

)()( FSTM

)(, FPZFis compact and convex. Extreme points are

set of trigonometric

polynomials f whose frequencies are in F.

Fk

kcdttff 21

0

22 |||)(|||||

.1||||),(:||closureweak 2 fFPff

Theorem 2 (Banach-Alaoglu) The set if probability

)()( TCTM with the weak*-topologymeasures.t

spectral envelope of .F

Spectral Envelopes

]),([ NMSg}:{],[ NkMZkNMF integer interval

Theorem 3 (Fejer-Riesz)

1

01)(,0,]),([ dttggMNNMPg

Corollary 1 )()(]),0([ TMZSS Proof First observe that for

every 1N the

Fejer kernel ]),0([|)1(| 2

021

NSeNK k

N

kN

hence )(0 TMK weakN

so for )(TM.

weakNN Kg

satisfies TtttNNtKN ),2/sin(/)2/)1sin(()1()( 1

Also ]),,([ NNPgN

.]),0([1)(,01

0NSgdttgg NNN

http://people.virginia.edu/~jlr5m/Papers/FejerRiesz.pdf

Spectral Envelopes

Corollary 2 ]),([ NMS is convex.

Lemma 1 ]),([ NMSg is an extreme point

all the roots of its Laurent polynomial .T

http://en.wikipedia.org/wiki/Choquet_theory

Theorem 4 (Choquet) Every

dteeetrr

rtsi

cos2112

21

0

2)(21

21

2 |)(2||)()||1(| 21

21

)||1/(||2,1||,|| 22 re is

]),([ NMSgrepresented by a measure on the extreme points.

is

Example 1

Feichtinger’s Conjecture for Exponentials

).(,|||||)(| 22 FPffdttfSt

is a Riesz Pair if such that

ZFTBorelSFS ),(),,(Definition

W. Lawton, Minimal sequences and the Kadison-Singer problem, http://arxiv.org/find/grp_math/1/au:+Lawton_W/0/1/0/all/0/1, November 30, 2009.

Bulletin Malaysian Mathematical Sciences Society (2) 33 (2), (2010) 169-176.

0

FCE )(TBorelSTheorem 5. (Lawton-Paulsen)

.),(..1n

j jj FZFSPR

)(TBorelS

syndetic. with ),(.... FFSPRPR

Verne Paulsen, Syndetic sets, paving, and the Feichtinger conjecture, http://arxiv.org/abs/1001.4510 January 25, 2010.

V. I. Pausen, A dynamical systems approach to the Kadison-Singer problem, Journal of Functional Analysis 255 (2008), 120-132.

Quadratic Optimization)(,)(ˆ)(ˆ)(ˆ|)(|

,

2 FPfdekfjkjfdttfFkj

SSt

Since

).(,|||||)(| 22 FPffdttfSt

the maximum 0 that satisfies

)][(eiginf FF RSR where

FR is the restriction ).()(: 22 ZFRF and

])(ˆ[),]([ jkkjS S ))((][ 2 ZBS

Theorem 6

is the Toeplitz matrix

has a bounded inverse.

),( FS is a Riesz Pair iff

))((][ 2 FBRSR FF

Numerical ExperimentsClearly the only candidate counterexamplesare Fat Cantor sets such as the set

]),(),(),[(\],[ 7223

7219

7219

7223

121

121

21

21 S

)()(limweak 21

21

21

21

21

11 nn xxxxn

S

constructed like Cantor’s ternary set but whose lengths of deleted open intervals are halved, so

where 2),32(, 121

247

1 nxx nnn hence

121 ).2cos()(ˆ

j jS ykk

Numerical Experiments

function A = cantor(N,M)% function A = cantor(N,M)y(1) = 7/24;for j = 2:M

y(j) = .5*(2^(-j-1)+3^(-j));endk = 0:N;A = 0.5*cos(2*pi*y(1)*k);for j = 2:M A = A.*cos(2*pi*y(j)*k);end

Background[KS59, Lem 5] A pure state on a max. s. adj. abelian subalgebra

iff uniquely extends to))(( 2 ZB Ais pavable. No for

[CA05] Feichtinger’s Conjecture Every bounded frame can be written as a finite union of Riesz sequences.

[KS59] R. Kadison and I. Singer, Extensions of pure states, AJM, 81(1959), 547-564.

[CA05] P. G. Casazza, O. Christiansen, A. Lindner and R. Vershynin, Frames and the Feichtinger conjecture, PAMS, (4)133(2005), 1025-1033.

B))(( 2 ZBB ).(ZAopen for]),1,0([LA

[CA06a, Thm 4.2] Yes answer to KSP equiv. to FC.

[CA06a] P. G. Casazza and J. Tremain, The Kadison-Singer problem in mathematics and engineering, PNAS, (7) 103 (2006), 2032-2039.

[CA06b] Multitude of equivalences.

[CA06b] P. G. Casazza, M. Fickus, J. Tremain, and E. Weber, The Kadison-Singer problem in mathematics and engineering, Contemp. Mat., 414, AMS, Providence, RI, 2006, pp. 299-355.

Lower and Upper Beurling

Densities of

|),(|minlim)( 1 kaaDRakk

and Separation

|),(|maxlim)( 1 kaaDRakk

Lower and Upper Asymptotic

|),(|mininflim)( 21 kkd

Rakk

|),(|minsuplim)( 21 kkd

Rakk

||min)( 2121

Z

Fat Cantor SetsSmith–Volterra–Cantor set (SVC) or the fat Cantor set is an example of a set of points on the real line R that is nowhere dense (in particular it contains no intervals), yet has positive measure. The Smith–Volterra–Cantor set is named after themathematicians Henry Smith, Vito Volterra and Georg Cantor.

http://en.wikipedia.org/wiki/File:Smith-Volterra-Cantor_set.svg

http://www.macalester.edu/~bressoud/talks/AlleghenyCollege/Wrestling.pdf

The Smith–Volterra–Cantor set is constructed by removing certain intervals from the unit interval [0, 1].

The process begins by removing the middle 1/4 from the interval [0, 1] to obtain                   

The following steps consist of removing subintervals of width 1/22n from the middle of each of the 2n−1 remaining

intervals. Then remove the intervals (5/32, 7/32) and (25/32, 27/32) to get                                 

Riesz Pairs.),/( ZZRTBorelS

))(( EPS

orTLhhPhPPS ),(||,||||||0.1 211

is a Riesz basis for its span.Definition ),( S is a Riesz Pair (RP) if

[LA09, Lem 1.1] RP),( S iff

W. Lawton, Minimal sequences and the Kadison-Singer problem, http://arxiv.org/find/grp_math/1/au:+Lawton_W/0/1/0/all/0/1, November 30, 2009.

Bull. Malysian Mathematical Society (2) 33 (2), (2010) 169-176.

PPS ;)(2 TL onto }).:{)(();()( 22 neEspanTLx inx

Sorth. proj. of

orTLhhhPhP ZS ),(||,||||||||||0.2 22\2

).(||,||||||0.3 2\3\\3 TLhhPhPP STSTZ

[LA09, Cor 1.1] RP),( S ).()( SD

H. J. Landau, Necessary density conditions for sampling and interpolation of certain entire functions, Acta Math. 117 (1967), 37-52.

Riesz Pairs

[MV74] RP),())(/1,( SSaa

[MV74] H. L. Montgomery and R. C. Vaughan, Hilbert's inequality, J. London Math. Soc., (2) 8 (1974), 73-82.

[CA01, Thm 2.2] TSRPmnZS nn

n },,,0{),( 11 (never the case if S is a fat Cantor set)

[CA01] P. G. Casazza, O. Christiansen, and N. Kalton, Frames of translates, Collect. Math., 52(2001), 35-54.

[BT87, Res. Inv. Thm.] RPSdS ),(0)(,

[BT87] J. Bourgain and L. Tzafriri, Invertibility of "large" submatrices with applications to the geometry of Banach spaces and harmonic analysis, Israel J. Mathematics, (2) 57 (1987),137-224.

[LA09, Thm 2.1] RPnot),(setBohr S

Stationary Setsa setDefinition For a discrete group G

is

[MV74]Feichtinger’s Conjecture holds forZ

SpaceHilbertGghg }:{).(,, 1

11 abhhhh

abba if

G stationary

stationary Bessel sets iff for every fatCantor set TS there exist a partition

.,...,1,RP),(1 mjSZ jm

J. Bourgain and L. Tzafriri, On a problem of Kadison and Singer, J. reine angew. Math., 420 (1991),1-43.

}:)({ 2 Znex inxS [BT91,Thm 4.1]

satisfies FC if .|||)(ˆ|)1,0( 2

kkZk

S

Syndetic Sets and Minimal Sequences

is syndetic if there exists a positive integerZ n with

.,...,2,1 Zn

Z1,0 is a minimal sequence if its orbit closure

These are core concepts in symbolic topological dynamics [GH55]

is a minimal closed shift-invariant set.

[GH55] W. H. Gottschalk and G. A. Hedlund, Topological Dynamics, Amer. Math. Soc., Providence, R. I., 1955.

Symbolic Dynamics Connection

1.

[LA09, Thm 1.1] These conditions are equivalent:There exists a partition

2.

3. min. seq. and

.,...,1,RP),(1 mjSZ jm

RP.),( S

V. Paulson, Syndetic Sets, Pavings and the Feichtinger Conjecture,

http://arxiv.org/abs/1001.4510 January 25, 2010.

[VP10] gives powerful extensions of this result.

V. Pauson, A dynamical systems approach to the Kadison-Singer problem, J. Functional Analysis, 225 (2008), 120-132.

.RP),(syndetic SZ Z

W. Lawton, Frames and the Kadison-Singer Problem, Wavelets and Appli- cations Conference, Euler Institute, St. Petersburg, Russia, June 14-20, 2009.

W. Lawton, Extending Pure States on C*-algebras and Feichtinger’s Con- jecture, Special Program on Operator Algebras, 5th Asian Mathematical Con- ference, Putra World Trade Center, Kuala Lumpur, Malaysia, June 22-26, 2009.

Power Spectral Measure

Theorem (Khinchin, Wiener, Kolmogorov)

Definition A function

exist.

k

kkkh hnhnR

)()(lim)( 2

1

is wide sense stationary if

Since

CZh :

k

kkkh hm

)(lim 2

1 and

22

21 )(limweak

k

k

xikk

h ehS

on ThR is positive definite the Bochner-Herglotz Theorem

such that

.),()(1

0

2 ZnxdSenR hinx

h implies there exists a positive measure

hS

W. Lawton, Riesz Pairs and Feichtinger’s Conjecture, International Conf. Mathematics and Applications, Twin Towers Hotel, Bangkok, Thailand,

December 17-19, 2009.

New Result Theorem If is

is wide sense stationary and

Zis a fat Cantor set and ifTS

0 there exists a closed set TS such that

)\( STm and TSS

then ),( S is not a RP.

Proof SySTyTSS )(

Define 1,)()( )(2

21

kexPk

k

yxi

kk

then dxxPS k

k

2|)(|lim

0

m

and for all

and

mdxxPkk

1

0

2|)(|lim

such that

Thue-Morse Minimal Sequence 010110011010011010010110 = b 101 bbb

The Thue–Morse sequence was first studied by Eugene Prouhet in 1851, who applied it to number theory. However, Prouhet did not mention the sequence explicitly; this was left to Axel Thue in 1906, who used it to found the study of combinatorics on words. The sequence was only brought to worldwide attention with the work of Marston Morse in 1921, when he applied it to differential geometry. The sequence has been discovered independently many times, not always by professional research mathematicians; for example, Max Euwe, a chess grandmaster and mathematics teacher, discovered it in 1929 in an application to chess: by using its cube-free property (see above), he showed how to circumvent a rule aimed at preventing infinitely protracted games by declaring repetition of moves a draw.

http://en.wikipedia.org/wiki/Thue%E2%80%93Morse_sequence

can be constructed for nonnegative n1. through substitutions 001,110

2. through concatenations 00|1 0|1|10 0|1|10|1001

3. 2mod ofexpansion 2 base in the s1' of # nbn 4. solution of Tower of Hanoi puzzle http://www.jstor.org/pss/2974693

Thue-Morse Spectral Measure

22

21 )(limweak

k

k

xikk

b ebS

1

0

241

041 )2(sin2limweak

n

k

k

nx

S. Kakutani, Strictly ergodic symbolic dynamical systems. In Proc. 6th Berkeley Symp. On Math. Stat. and Prob., eds. Le Cam L. M., Neyman J. and Scott E. El., UC Press, 1972, pp. 319-326.

can be represented using a Riesz product

[KA72] Theorem 2nd term is purely singular continuous and

has dense support.

Corollary Let ,...}7,4,2,1,2,3,5,8{...,)(support bTM

For every 0 there exists a fat Cantor set S such that

1)( Measure Lebesgue S and ),( TMS is not a RP.

Volterra Iteration

x n

k

k

ndyyxF

0

1

0

2 )2(sin2limweak)(

that approximates the cumulative distribution

is given by

21

2

0

221

1 0,)()]2/(sin2[)( xyFdyxFx

nn

1),1(1)( 21

11 xxFxF nn

and is a weak contraction with respect to the total variation norm [BA08] and hence it converges uniformly to

M. Baake and U. Grimm, The singular continuous diffraction measure of the Thue-Morse chain, J. Phys. A: Math. Theor. 41 (2008) 422001 (6pp) , arXiv:0809.0580v2

,10,)(1 xxxF

.F

MATLAB CODEfunction [x,F] = Volterra(log2n,iter)% function [x,F] = Volterra(log2n,iter)%n = 2^log2n;dx = 1/n;x = 0:dx:1-dx;S = sin(pi*x/2).^2;F = x;for k = 1:iter

dF = F - [0 F(1:n-1)];P = S.*dF;I = cumsum(P);F(1:n/2) = I(1:2:n);F(n/2+1:n) = 1 - F(n/2:-1:1);

end

Thue-Morse Distribution 20 iterations

Thue-Morse Spectral Measure

Spline Approximation AlgorithmIs obtained by replacing

is given by

21

2

021

1 0,)()()( xyFdyAxFx

an

an

1),1(1)( 21

11 xxFxF an

an

also converges uniformly to an approximation

10,)(1 xxxF a

)1,[on 1),,0[on 1)()2/(sin2 21

21

21

212 yAy

aF to .F

Spline Approx. Distribution (20 iterations)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Spline Approx. Spectral Measure

Distribution Comparison

Binary Tree Model

)(2 1adF

0

21

21 1,1 ba

21

1

041

41

21

)(4 2adF a b

)(8 3adF

0 81

81

41

aa ab41

83

83

21

bb ba

babbaabbabbbabaabbaabaaadF a )(16 48536.01464.08536.09749.41464.08536.01464.00251.0

Binomial Approximation

),0( 21

mnmba

For every and 2n the intervals that

contribute 21

1 )intervals ofunion (andF

are those with m a’s and (n-m) b’s with hence

4660650.303390072lnln2ln

2lnln2/)(lnln

cn

mbb

bnb

so the fraction of these dyadic intervals is

dtttcn

ncnn

k

n cncnncn

k

n

21

0

][1][][

0)1(

][])[(2

)2741940.03865546exp())(exp())/][((exp2 2212

21 ncnncnnn

Hausdorff-Besicovitch Dimension d]1,0[ dimensional H. content of a subset ]1,0[S

j j

jdi

dH ISLSC :inf)(

0)(:0inf)(dim SCdS dHH

S. Besicovitch (1929). "On Linear Sets of Points of Fractional Dimensions". Mathematische Annalen 101 (1929). S. Besicovitch; H. D. Ursell (1937). "Sets of Fractional Dimensions". J. London Mathematical Society 12 (1937).F. Hausdorff (March 1919). "Dimension und äußeres Maß". Mathematische Annalen 79 (1–2): 157–179.

Theorem For the approximate support S of adFndn

n

dH nSC

2)2741940.03865546exp(2lim)(

])2ln2741940.038655462(ln[explim dnn

therefore 5598940.94423195)(dim SH

Thickness of Cantor Sets

101100 ,, AACAOAAAC

[AS99] S. Astels, Cantor sets and numbers with restricted partial quotients, TAMS, (1)352(1999), 133-170.

Thickness

111101010000 , AOAAAOAA

111001002 AAAAC 0

j

jCC

||

||,

||

||mininf)(

10

O

A

O

AC

Ordered Derivation

|||||,||| 10 OOOO

))(1/()()( CCC

[AS99] Thm 2.4 Let kCC ,...,1

kCCCC 111 1)()( contains an interval.

be Cantor sets. Then

)})()(,1min{/11ln(/2ln)(dim 111 CCCC kH

Research Questions1.Clearly fat Cantor sets have Hausdorff dim =1 and thickness = 1. What are these parameters for approximate supports of spectral measures of the Thue-Morse and related sequences?

3. How are these parameters related to the Riesz properties of pairs ?),( S

M. Keane, Generalized Morse sequences, Z.

Wahrscheinlichkeitstheorie verw. Geb. 10(1968),335-353

4. What happens for gen. Morse seq. [KE68]?

2. How are these properties related to multifractal properties of the TM spectral measure [BA06]?

Zai-Qiao Bai, Multifractal analysis of the spectral measure of the

Thue-Morse sequence: a periodic orbit approach, J. Phys. A: Math. Gen.

39(2006) 10959-10973.