1. Spectral Envelopes, Riesz Pairs, and Feichtinger’s Conjecture University of Newcastle, AUSTRALIA September 23, 2010 Wayne Lawton Department of Mathematics National University of Singapore matwml@nus.edu.sg http://www.math.nus.edu.sg/~matwml

2. Frames and Riesz Sets e.g. Consider a complex Hilbert space Definition S is a frame ; Riesz set if and give Parseval frames = http://en.wikipedia.org/wiki/POVMhttp://en.wikipedia.org/wiki/POVM ; give orthonormal sets.

3. Relation to Kadison-Singer [KS59, Lem 5] A pure state on a max. s. adj. abelian subalgebra iffuniquely extends to is pavable. No for [CA05] Feichtinger’s Conjecture Every frame (with norms of its elements bounded below) is a finite union of Riesz sets. [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. open for [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.

4. F.C. for Frames of Translates give OS give frames but only However are OS are RS

5. Fourier Tricks for the Upper Frame Bound where the Grammian

6. Fourier Tricks for the Riesz Bounds where thereforeis a Riesz set if and only if for almost all

7. Translations by Arithmetic Sequences where sois a Riesz set if and only if for almost all H. Halpern, V. Kaftal, and G. Weiss,The relative Dixmier property in discrete crossed products, J. Funct. Anal. 69 (1986), 121-140. Matrix pavings and Laurent operators, J. Operator Theory 16#2(1986), 355-374. HKW86 Ifis Riemann integrable then satisfies Feichtinger’s conjecture with each RS of the formand approx. orthogonal. [CA01] P. G. Casazza, O. Christiansen, and N. Kalton, Frames of translates, Collect. Math., 52(2001), 35-54. CCK01 Fails ifwitha Cantor set.

8. Feichtinger’s Conjecture for Exponentials is a Riesz Pair if such that satisfies FCE if FCE : Everysatisfies FCE or equivalently every Cantor setsatisfies FCE. FCE  FC for frames of translates.

9. Quadratic Optimization Since the maximum that satisfies where is the restrictionand Theorem is the Toeplitz matrix has a bounded inverse. iff

10. Equivalences W. Lawton, Minimal sequences and the Kadison-Singer problem, http://arxiv.org/find/grp_math/1/au:+Lawton_W/0/1/0/all/0/1http://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. (L,Lemma 1.1) is a Riesz pair is a Riesz basis for is a frame for can be ‘robustly reconstructed from samples

11. Lower and Upper Beurling Properties of Integer Sets and Separation Lower and Upper Asymptotic

12. Characterizing Riesz Pairs [MV74] Corollary 2 [MV74] H. L. Montgomery and R. C. Vaughan, Hilbert's inequality, J. London Math. Soc., (2) 8 (1974), 73-82. [BT87,SS09] Res. Inv. Thm. [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 satisfies FCE (e.g by removing open intervals with exp. decr. lengths) [LT91] J. Bourgain and L. Tzafriri, On a problem of Kadison and Singer, J. reine angew. Math., {\bf 420}(1991),1-43. [SS09] D. A. Spielman and N. Srivastava, An elementary proof of the restricted invertibility theorem, arXiv:0911.1114v1 [math.FA] 5 Nov 2009. [LA09] Corollary 1.1 [BT91] M. Ledoux and M. Talagrand, Probability in Banach Spaces, 15.4 Invertibility of Submatrices, pp. 434-437. Springer, Heidelberg, 1991.

13. Syndetic Sets and FCE is syndetic if W. Lawton, Minimal sequences and the Kadison-Singer problem, http://arxiv.org/find/grp_math/1/au:+Lawton_W/0/1/0/all/0/1http://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. Theorem (L,Paulsen) Verne Paulsen, Syndetic sets, paving, and the Feichtinger conjecture, http://arxiv.org/abs/1001.4510http://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. satisfies FCE if and only if there exists a syndetic such thatis a Riesz pair.

14. Research Problem 1. The Cantor set constructed like Cantor’s ternary set but whose lengths of deleted open intervals are halved, so where hence and Riesz pair with syndetic COMPUTE IT !

18. Polynomials Laurent trigonometric Jensen

19. Spectral Envelopes compact and convex. Extreme points are set of trigonometric polynomials f whose frequencies are in F. (Banach-Alaoglu) The set if probability measures with the weak*-topology is spectral envelope of

20. Symbolic Dynamics There is a bijection between integer subsets and points in the has the product topology and the shift homeomorphism Bebutov (symbolic) dynamical system M. V. Bebutov, On dynamical systems in the space of continuous functions, Bull. Mos. Gos. Univ. Mat. 2 (1940). Orbit closures are closed shift invariant subsets where A pointis recurrent if for every open there exists a nonzerowith

21. Research Problem 2. Theorem If is recurrent thenTheorem If Proof Follows from the Riemann-Lebesque lemma. is convex. then Furthermore, ifis nonempty thenis infinite and the set of extreme points consists of limits of squared moduli of polynomials whose coefficients converge uniformly to zero. What isand how is it related to the dynamical and ergodic properties of the shift dynamical system

22. Sample Result for RP 2. Theorem. Letbe a shift invariant ergodic measure on andThen the positive definite function is the Fourier transform of Example If is wide sense stationary then

23. Spectral Envelopes integer interval Fejer-Riesz Corollary Proof First observe that for everythe Fejer kernel henceso for satisfies Also http://people.virginia.edu/~jlr5m/Papers/FejerRiesz.pdf

24. Spectral Envelopes Corollaryis convex. Lemma http://en.wikipedia.org/wiki/Choquet_theory Choquet Every represented by a measure on the extreme points. is Example

25. Syndetic Sets and Minimal Sequences is a minimal sequence if is a minimal closed shift-invariant set. [GH55] W. H. Gottschalk and G. A. Hedlund, Topological Dynamics, Amer. Math. Soc., Providence, R. I., 1955. [G46] W. H. Gottschalk, Almost periodic points with respect to transformation semigroups, Annals of Mathematics, 47 (1946), 762-766. [G46]is a minimal sequence iff for every open the set is syndetic. [F81] Theorem 1.23 Ifthen some contains minimal sequencepiecewise syndetic. [F81] H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, New Jersey, 1981.

26. Thue-Morse Minimal 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 wasnumber theory left to Axel Thue in 1906, who used it to found the study of combinatorics on words.Axel Thuecombinatorics on words The sequence was only brought to worldwide attention with the work of Marston MorseMarston Morse in 1921, when he applied it to differential geometry. The sequence has been discovereddifferential geometry independently many times, not always by professional research mathematicians; for example, Max Euwe, a chess grandmaster and mathematics teacher, discovered it inMax Euwechess grandmasterteacher 1929 in an application to chess: by using its cube-free property (see above), he showedchess 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 isconstructed for 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

27. Thue-Morse Spectral Measure S. Kakutani, Strictly ergodic symbolic dynamical systems. In Proc. 6 th 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] 2 nd term is purely singular continuous with dense support.

28. Thue-Morse Spectral Measure

30. 1 2 4 7 8 11 13 14 16 19 21 22 25 26 28 31 32 35 37 38 41 42 44 47 49 50 52 55 56 59 61 62 Morse F=

31. Bohr Minimal Sets and Sequences Let and define and let and define the Bohr set Theoremis a minimal sequence and is positive definite on

32. Research Problem 3. Group Theory Z  discrete group D, T  extreme pos. def. functions on D that = 1 at identity, or T  compact group G and Z  matrix entries of irred. representations of G. Generalize