Riesz Pairs and Feichtinger’s Conjecture

1. Riesz Pairs and Feichtinger’s Conjecture INTERNATIONAL CONFERENCE IN MATHEMATICS AND APPLICATIONS (ICMA - MU 2009) Wayne Lawton Department of Mathematics National University of Singapore matwml@nus.edu.sg http://www.math.nus.edu.sg/~matwml

2. Titles Volterra Iteration MATLAB Code Thue-Morse Distribution Thue-Morse Spec. Meas. Spline Approx. Algorithm Spline Approx. Distribution Spline Approx. Spec. Meas. Distribution Comparison Binary Tree Model Binomial Approximation Hausdorff-Besicovitch Dim. Thickness of Cantor Sets Research Questions References Background Equivalences Subject Syndetic Sets Min. Seq. Symbolic Dynamics Objectives Densities Fat Cantor Sets Known Results Power Spectral Measure New Result Thue-Morse Min. Seq. Tower of Hanoi Thue-Morse Spec. Meas.

3. Background Recently there has been considerable interest in two deep problems that arose from very separate areas of mathematics. Kadison-Singer Problem (KSP): Does every pure state on the -subalgebra admit a unique extension to arose in the area of operator algebras and has remained unsolved since 1959 [KS59]. Feichtinger’s Conjecture (FC): Every bounded frame can be written as a finite union of Riesz sequences. arose from Feichtinger's work in the area of signal processing involving time-frequency analysis and has remained unsolved since it was formally stated in the literature in 2005 [CA05]. [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, Frames and the Feichtinger conjecture, PAMS, (4)133(2005), 1025-1033.

4. Equivalences Casazza and Tremain proved ([CA06b], Thm 4.2) that a yes answer to the KSP is equivalent to FC. Casazza, Fickus, Tremain, and Weber [CA06a] explained numerous other equivalences. [CA06b] P. G. Casazza and J. Tremain, The Kadison-Singer problem in mathematics and engineering, PNAS, (7) 103 (2006), 2032-2039. [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.

5. Subject of this talk is the following special case of FC: Feichtinger’s Conjecture for Exponentials (FCE): For every non-trivial measurable set the sequence is a finite union of Riesz sequences*. *If is a Riesz sequence if there exists such that every trigonometric polynomial (with frequencies in ) satisfies

6. Syndetic Sets and Minimal Sequences is syndetic if there exists a positive integer with is a minimal sequence if its orbit closure is a minimal closed shift-invariant set. These are core concepts in symbolic topological dynamics [GH55] [GH55] W. H. Gottschalk and G. A. Hedlund, Topological Dynamics, Amer. Math. Soc., Providence, R. I., 1955.

7. Theorem 1.1 [LA09] For measurable the Symbolic Dynamics Connection following conditions are equivalent: is a finite union of Riesz sequences. 1. such that There exists a syndetic set 2. is a Riesz sequence. such that There exists a nonempty set 3. 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

8. Objectives Near Term: Characterize Riesz pairs (pairs such that is a Riesz basis) Long Term: Contribute to the understanding of FCE and hopefully to FC and the KS problem.

9. Densities Lower and Upper Beurling Lower and Upper Asymptotic and Separation

10. Fat Cantor Sets Smith–Volterra–Cantor set (SVC) or the fat Cantor set is an example of a set of points on the real lineR that is nowhere dense (in particular it contains no intervals), yet has positive measure. The Smith–Volterra–Cantor set is named after the mathematiciansHenry Smith, Vito Volterra and Georg Cantor. 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 http://en.wikipedia.org/wiki/File:Smith-Volterra-Cantor_set.svg http://www.macalester.edu/~bressoud/talks/AlleghenyCollege/Wrestling.pdf

11. Known Results [LA09] Corollary 1.1 [MV74] Corollary 2 [CA01] Theorem 2.2 (never the case if S is a Cantor set) [BT87] Res. Inv. Thm. [BT91] Theorem 4.1 (occurs if S is a boring fat Cantor set) [LA09] Theorem 2.1 [MV74] H. L. Montgomery and R. C. Vaughan, Hilbert's inequality, J. London Math. Soc., (2) 8 (1974), 73-82. [CA01] P. G. Casazza, O. Christiansen, and N. Kalton, Frames of translates, Collect. Math., 52(2001), 35-54. [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] J. Bourgain and L. Tzafriri, On a problem of Kadison and Singer, J. reine angew. Math., {\bf 420}(1991),1-43.

12. is wide sense stationary if Definition A function Power Spectral Measure and exist. Since is positive definite the Bochner-Herglotz Theorem implies there exists a positive measure on such that Theorem (Khinchin, Wiener, Kolmogorov)

13. is Theorem If is a fat Cantor set and if New Result such that is wide sense stationary and and for all there exists a closed set such that and then is not a RP. Proof Define then and

14. Thue-Morse Minimal Sequence can be constructed for nonnegative 1. through substitutions 001,110 2. through concatenations 00|1 0|1|10  0|1|10|1001  3. 4. solution of Tower of Hanoi puzzle http://www.jstor.org/pss/2974693 http://en.wikipedia.org/wiki/Thue%E2%80%93Morse_sequence 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.

16. can be represented using a Riesz product Thue-Morse Spectral Measure [KA72] Theorem 2nd term is purely singular continuous and has dense support. Corollary Let For every there exists a fat Cantor set such that and is not a RP. [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.

17. that approximates the cumulative distribution Volterra Iteration is given by 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

18. MATLAB CODE function [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

19. Thue-Morse Distribution 20 iterations

20. Thue-Morse Spectral Measure

21. Is obtained by replacing Spline Approximation Algorithm is given by also converges uniformly to an approximation to

22. Spline Approx. Distribution (20 iterations)

23. Spline Approx. Spectral Measure

24. Distribution Comparison

25. Binary Tree Model

26. For every and the intervals that Binomial Approximation contribute hence are those with m a’s and (n-m) b’s with so the fraction of these dyadic intervals is

27. dimensional H. content of a subset Hausdorff-Besicovitch Dimension Theorem For the approximate support of therefore S. Besicovitch (1929). "On Linear Sets of Points of Fractional Dimensions". Mathematische Annalen101 (1929). S. Besicovitch; H. D. Ursell (1937). "Sets of Fractional Dimensions". J. London Mathematical Society12 (1937). F. Hausdorff (March 1919). "Dimension und äußeres Maß". Mathematische Annalen79 (1–2): 157–179.

28. Ordered Derivation Thickness of Cantor Sets Thickness [AS99] Thm 2.4 Let be Cantor sets. Then contains an interval. [AS99] S. Astels, Cantor sets and numbers with restricted partial quotients, TAMS, (1)352(1999), 133-170.

29. Research Questions • 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? 2. How are these properties related to multifractal properties of the TM spectral measure [BA06]? 3. How are these parameters related to the Riesz properties of pairs 4. What happens for gen. Morse seq.[KE68]? [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. [KE68] M. Keane, Generalized Morse sequences, Z. Wahrscheinlichkeitstheorie verw. Geb. 10(1968),335-353

30. J. Anderson, Extreme points in sets of positive linear maps on B(H), J. Func. Anal. 31(1979), 195-217. References M. V. Bebutov, On dynamical systems in the space of continuous functions, Bull. Mos. Gos. Univ. Mat. 2 (1940). H. Bohr, Zur Theorie der fastperiodischen Funktionen I,II,III. Acta Math. 45(1925),29-127;46(1925),101-214;47(1926),237-281 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