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Spectral Envelopes, Riesz Pairs, and Feichtinger’s Conjecture

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

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Spectral Envelopes, Riesz Pairs, and Feichtinger’s Conjecture

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

  2. Frames and Riesz Sets Consider a complex Hilbert space e.g. and Definition S is a frame ; Riesz set if give Parseval frames = http://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 uniquely extends to iff is pavable. No for open for [CA05] Feichtinger’s Conjecture Every frame (with norms of its elements bounded below) is a finite union of Riesz sets. [CA06a, Thm 4.2] Yes answer to KSP equiv. to FC. [CA06b] Multitude of equivalences. [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. [CA06a] P. G. Casazza and J. Tremain, The Kadison-Singer problem in mathematics and engineering, PNAS, (7) 103 (2006), 2032-2039. [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 frames but only give OS However are OS are RS

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

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

  7. Translations by Arithmetic Sequences where so is a Riesz set if and only if for almost all HKW86 If is Riemann integrable then satisfies Feichtinger’s conjecture with each RS of the form and approx. orthogonal. CCK01 Fails if with a Cantor set. 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. [CA01] P. G. Casazza, O. Christiansen, and N. Kalton, Frames of translates, Collect. Math., 52(2001), 35-54.

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

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

  10. (L,Lemma 1.1) is a Riesz pair is a Riesz basis for Equivalences is a frame for can be ‘robustly reconstructed from samples 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.

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

  12. Characterizing Riesz Pairs [LA09] Corollary 1.1 [MV74] Corollary 2 [BT87,SS09] Res. Inv. Thm. [BT91] Theorem 4.1 satisfies FCE (e.g by removing open intervals with exp. decr. lengths) [MV74] H. L. Montgomery and R. C. Vaughan, Hilbert's inequality, J. London Math. Soc., (2) 8 (1974), 73-82. [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. [LT91] J. Bourgain and L. Tzafriri, On a problem of Kadison and Singer, J. reine angew. Math., {\bf 420}(1991),1-43. [BT91] M. Ledoux and M. Talagrand, Probability in Banach Spaces, 15.4 Invertibility of Submatrices, pp. 434-437. Springer, Heidelberg, 1991. [SS09] D. A. Spielman and N. Srivastava, An elementary proof of the restricted invertibility theorem, arXiv:0911.1114v1 [math.FA] 5 Nov 2009.

  13. is syndetic if Theorem (L,Paulsen) satisfies FCE Syndetic Sets and FCE if and only if there exists a syndetic such that is a Riesz pair. 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. 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.

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

  15. Polynomials Laurent trigonometric Jensen

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

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

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

  19. Sample Result for RP 2. Theorem. Let be a shift invariant ergodic measure on and Then the positive definite function is the Fourier transform of Example If is wide sense stationary then

  20. Spectral Envelopes integer interval Fejer-Riesz Corollary Proof First observe that for every the Fejer kernel satisfies hence so for Also http://people.virginia.edu/~jlr5m/Papers/FejerRiesz.pdf

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

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

  23. isconstructed for 1. through substitutions 001,110 Thue-Morse Minimal Sequence 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.

  24. can be represented using a Riesz product Thue-Morse Spectral Measure [KA72] 2nd term is purely singular continuous with dense support. 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.

  25. Thue-Morse Spectral Measure

  26. Morse F= 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

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

  28. Research Problem 3. Generalize 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.

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