1. Spectral Envelopes of Integer Subsets 2 nd Asian Conference on Nonlinear Analysis and Optimization Patong beach, Phuket, THAILAND September 9-12, 2010 Wayne Lawton Department of Mathematics National University of Singapore matwml@nus.edu.sg http://www.math.nus.edu.sg/~matwml

2. Characters integer, real, circle group (real) characters (complex exponential functions) complex numbers circle group (complex)

3. Polynomials finite subset Laurent polynomial trigonometric polynomial whose frequencies are in F Theorem 1 (Jensen)

4. Spectral Envelopes is compact and convex. Extreme points are set of trigonometric polynomials f whose frequencies are in F. Theorem 2 (Banach-Alaoglu) The set if probability with the weak*-topologymeasures spectral envelope of

5. Spectral Envelopes integer interval Theorem 3 (Fejer-Riesz) Corollary 1 Proof First observe that for everythe Fejer kernel henceso for satisfies Also http://people.virginia.edu/~jlr5m/Papers/FejerRiesz.pdf

6. Spectral Envelopes Corollary 2is convex. Lemma 1 is an extreme point all the roots of its Laurent polynomial http://en.wikipedia.org/wiki/Choquet_theory Theorem 4 (Choquet) Every represented by a measure on the extreme points. is Example 1

7. Feichtinger’s Conjecture for Exponentials is a Riesz Pair if such that Definition W. Lawton, Minimal sequences and the Kadison-Singer problem, http://arxiv.org/find/grp_math/1/au:+Lawton_W/0/1/0/all/0/1http://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. FCE Theorem 5. (Lawton-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.

8. Quadratic Optimization Since the maximum that satisfies where is the restrictionand Theorem 6 is the Toeplitz matrix has a bounded inverse. is a Riesz Pair iff

9. Numerical Experiments Clearly the only candidate counterexamples are Fat Cantor sets such as the set constructed like Cantor’s ternary set but whose lengths of deleted open intervals are halved, so wherehence

10. 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)); end k = 0:N; A = 0.5*cos(2*pi*y(1)*k); for j = 2:M A = A.*cos(2*pi*y(j)*k); end

15. Background [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 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. 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.

16. Lower and Upper Beurling Densities of and Separation Lower and Upper Asymptotic

17. Fat Cantor Sets Smith–Volterra–Cantor set (SVC) or the fat Cantor set is an example of a set of points on the real line R that is nowherereal linenowhere densedense (in particular it contains no intervals), yet has positiveintervals measuremeasure. The Smith–Volterra–Cantor set is named after the mathematiciansmathematicians Henry Smith, Vito Volterra and Georg Cantor.Henry SmithVito VolterraGeorg 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].unit interval 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 1/2 2n from the middle of each of the 2 n−1 remaining intervals. Then remove the intervals (5/32, 7/32) and (25/32, 27/32) to get

18. Riesz Pairs is a Riesz basis for its span. Definitionis a Riesz Pair (RP) if [LA09, Lem 1.1] iff W. Lawton, Minimal sequences and the Kadison-Singer problem, http://arxiv.org/find/grp_math/1/au:+Lawton_W/0/1/0/all/0/1http://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. onto orth. proj. of [LA09, Cor 1.1] H. J. Landau, Necessary density conditions for sampling and interpolation of certain entire functions, Acta Math. 117 (1967), 37-52.

19. Riesz Pairs [MV74] [MV74] H. L. Montgomery and R. C. Vaughan, Hilbert's inequality, J. London Math. Soc., (2) 8 (1974), 73-82. [CA01, Thm 2.2] (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.] [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]

20. Stationary Sets a set Definition For a discrete group is [MV74]  Feichtinger’s Conjecture holds for if stationary stationary Bessel sets iff for every fat Cantor setthere exist a partition J. Bourgain and L. Tzafriri, On a problem of Kadison and Singer, J. reine angew. Math., 420 (1991),1-43. [BT91,Thm 4.1] satisfies FC if

21. Syndetic Sets and Minimal Sequences is syndetic if there exists a positive integerwith 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.

22. Symbolic Dynamics Connection 1. [LA09, Thm 1.1] These conditions are equivalent: There exists a partition 2. 3. min. seq. and V. Paulson, Syndetic Sets, Pavings and the Feichtinger Conjecture, http://arxiv.org/abs/1001.4510 January 25, 2010. http://arxiv.org/abs/1001.4510 [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. 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, 5 th Asian Mathematical Con- ference, Putra World Trade Center, Kuala Lumpur, Malaysia, June 22-26, 2009.

23. Power Spectral Measure Theorem (Khinchin, Wiener, Kolmogorov) Definition A function exist. is wide sense stationary if Since and on is positive definite the Bochner-Herglotz Theorem such thatimplies there exists a positive measure W. Lawton, Riesz Pairs and Feichtinger’s Conjecture, International Conf. Mathematics and Applications, Twin Towers Hotel, Bangkok, Thailand, December 17-19, 2009.

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

25. 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 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

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] Theorem 2 nd term is purely singular continuous and has dense support. Corollary Let For everythere exists a fat Cantor setsuch that andis not a RP.

28. Volterra Iteration that approximates the cumulative distribution 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:0809.0580v2arXiv:0809.0580v2

29. 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

30. Thue-Morse Distribution 20 iterations

31. Thue-Morse Spectral Measure

32. Spline Approximation Algorithm Is obtained by replacing is given by also converges uniformly to an approximationto

33. Spline Approx. Distribution (20 iterations)

34. Spline Approx. Spectral Measure

35. Distribution Comparison

36. Binary Tree Model

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

38. Hausdorff-Besicovitch Dimension dimensional H. content of a subset S. BesicovitchS. Besicovitch (1929). "On Linear Sets of Points of Fractional Dimensions". Mathematische Annalen 101 (1929).Mathematische Annalen S. BesicovitchS. Besicovitch; H. D. Ursell (1937). "Sets of Fractional Dimensions". J. London Mathematical Society 12 (1937).H. D. Ursell F. HausdorffF. Hausdorff (March 1919). "Dimension und äußeres Maß". Mathematische Annalen 79 (1–2): 157–179. Theorem For the approximate supportof therefore

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

40. Research Questions 1.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 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.