A New Strategy for Feichtinger’s Conjecture for Stationary Frames

1. A New Strategy for Feichtinger’s Conjecture for Stationary Frames Wayne Lawton Department of Mathematics National University of Singapore matwml@nus.edu.sg http://www.math.nus.edu.sg/~matwml http://arxiv.org/find/math/1/au:+Lawton_W/0/1/0/all/0/1 Applied & Computational Mathematics Seminar National University of Singapore 4PM, 20 January 2010      S16 Tutorial Room

2. is specified set of (integer) frequencies Trigonometric Polynomials These polynomials describe functions RC having period 1. Physical Models amplitude, freq. = k component signal amplitude, time = t k-th convolution-filter coefficient filter response, freq. = t k-th phased array amplitude beam amplitude, position = t k-th time series autocorr. coef. power spectrum, freq. = t

3. and is measurable, Riesz-Pairs Definition If then is a Riesz-pair if is a Riesz basis, this means that there exists such that Definition will denote the lub that satisfy the inequality above, thus

4. RP for every Examples RP if the separation [MV74] RP and asymptotic [BT87] density NRP if NRP if the upper Beurling density [LA09] NRP if and is nowhere dense. NRP if is a Bohr minimal sequence. [LA09] H. L. Montgomery and R. C. Vaughan, Hilbert's inequality, J.London Math.Soc., (2) 8 (1974), 73-82. 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. W. Lawton, Minimal Sequences and the Kadison-Singer Problem, accepted BMMSS

5. 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

6. Robust Signal Recovery Applications known set of possible non-zero frequency components set over which the signal is measured RP Signal can be robustly recovered iff Beam Nulling known set of transmitter locations set of locations where beam should be undetectable Beam can be nulled iff NRP

7. Given where Signal Recovery the convolution property for Fourier series gives

8. Two Celebrities 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.

9. 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.

10. Feichtinger’s Conjecture for Stationary Frames is equivalent to the following special case of FC: Feichtinger’s Conjecture for Exponentials (FCE): For every measurable set with where are RP. [BT91] Theorem 4.1 Feichtingers conjecture holds if [BT91] This condition holds for some Cantor sets [LA09] This condition does not hold for all Cantor sets [BT91] J. Bourgain and L. Tzafriri, On a problem of Kadison and Singer, J. reine angew. Math., {\bf 420}(1991),1-43.

11. 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. Core concepts in symbolic topological dynamics [G46], [GH55] [G46] W. H. Gottschalk, Almost periodic points with respect to transformation semigroups, Annals of Math., 47, (1946), 762-766. [GH55] W. H. Gottschalk and G. A. Hedlund, Topological Dynamics, Amer. Math. Soc., Providence, R. I., 1955.

12. Theorem 1.1 [LA09] For measurable the Symbolic Dynamics Connection following conditions are equivalent: is a finite union of Riesz seq. 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