Riesz Pairs and Feichtinger’s Conjecture

INTERNATIONAL CONFERENCE IN MATHEMATICS AND APPLICATIONS

(ICMA - MU 2009)

Wayne Lawton

Department of Mathematics

National University of Singaporematwml@nus.edu.sg

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

TitlesBackgroundEquivalencesSubjectSyndetic Sets Min. Seq.Symbolic DynamicsObjectivesDensitiesFat Cantor SetsKnown ResultsPower Spectral MeasureNew ResultThue-Morse Min. Seq.Tower of HanoiThue-Morse Spec. Meas.

Volterra IterationMATLAB CodeThue-Morse DistributionThue-Morse Spec. Meas.Spline Approx. AlgorithmSpline Approx. DistributionSpline Approx. Spec. Meas.Distribution ComparisonBinary Tree ModelBinomial ApproximationHausdorff-Besicovitch Dim.Thickness of Cantor SetsResearch QuestionsReferences

BackgroundRecently there has been considerable interest in two deep problems that arose from very separate areas of mathematics.

arose from Feichtinger's work in the area of signal processing involving time-frequency analysis and has remained unsolvedsince it was formally stated in the literature in 2005 [CA05].

Kadison-Singer Problem (KSP): Does every pure state on the

C -subalgebra )(Z admit a unique extension to ?))(( 2 ZB arose in the area of operator algebras and has remainedunsolved since 1959 [KS59].

Feichtinger’s Conjecture (FC): Every bounded frame canbe written as a finite union of Riesz sequences.

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

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

Equivalences

Casazza and Tremain proved ([CA06b], Thm 4.2)

that a yes answer to the KSP is equivalent to FC.

[CA06b] P. G. Casazza and J. Tremain, The Kadison-Singer problem in mathematics and engineering, PNAS, (7) 103 (2006), 2032-2039.

Casazza, Fickus, Tremain, and Weber [CA06a]

explained numerous other equivalences.

[CA06a] 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.

Subject

Feichtinger’s Conjecture for Exponentials (FCE):

of this talk is the following special case of FC:

For every non-trivial measurable set ZRTS /the sequence

is a finite union of Riesz sequences*.

ZktkietZSB

S 2)(),(

*If ),(, SBZ is a Riesz sequence if there exists0 such that every trigonometric polynomial

)(with frequencies intki

k k ectP 2)(

satisfies

k kTScdttPdttP 222

||)()(

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 Connectionthe

),( ZSBZ

1.

following conditions are equivalent:Theorem 1.1 [LA09] For measurable TS

is a finite union of Riesz sequences.

2. There exists a syndetic set

is a Riesz sequence.

such that

),( ZSB3. There exists a nonempty set Z such that

),( ZSB is a minimal sequence and

is a Riesz sequence.

[LA09] Minimal Sequences and the Kadison-Singer Problem, accepted BMMSS

http://arxiv.org/find/math/1/au:+Lawton_W/0/1/0/all/0/1

Objectives

Near Term: Characterize Riesz pairs

),( SB),( S

(pairs such that is a Riesz basis)

Long Term: Contribute to the understanding of FCE

and hopefully to FC and the KS problem.

Lower and Upper Beurling

Densities

|),(|minlim)( 1 kaaDRakk

and Separation

|),(|maxlim)( 1 kaaDRakk

Lower and Upper Asymptotic

|),(|mininflim)( 21 kkd

Rakk

|),(|minsuplim)( 21 kkd

Rakk

||min)( 2121

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                                 

Known Results

[LA09] Corollary 1.1 meas(S)DRPS )(),([MV74] Corollary 2 RPSSaa ),())(/1,(

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

[CA01] Theorem 2.2 TSRPmnZS nn

n },,,0{),( 11 (never the case if S is a 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.

[BT91] Theorem 4.1

|||)(ˆ|),1,0( 2 kkZk

S

RPSsyn ),( (occurs if S is a boring fat Cantor set)

[BT91] J. Bourgain and L. Tzafriri, On a problem of Kadison and Singer, J. reine angew. Math., {\bf 420}(1991),1-43.

[LA09] Theorem 2.1 RPS not),(setBohr

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

1and

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

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

)\( STS 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

[KA72] 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

[KE68] 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]?

[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.

ReferencesJ. Anderson, Extreme points in sets of positive linear maps on B(H), J. Func. Anal. 31(1979), 195-217.

H. Bohr, Zur Theorie der fastperiodischen Funktionen I,II,III. Acta Math. 45(1925),29-127;46(1925),101-214;47(1926),237-281

M. V. Bebutov, On dynamical systems in the space of continuous functions, Bull. Mos. Gos. Univ. Mat. 2 (1940).

O. Christenson, An Introduction to Frames and Riesz Bases, Birkhauser, 2003.H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton Univ. Press, 1981.H. Halpern, V. Kaftal, and G. Weiss,The relative Dixmier property in discrete crossed products, J. Funct. Anal. 69 (1986), 121-140.H. Halpern, V. Kaftal, and G. Weiss, Matrix pavings and Laurent operators, J. Operator Theory 16#2(1986), 355-374.N. Weaver, The Kadison-Singer problem in discrepancy theory, preprint