A New Strategy for Feichtinger’s Conjecture for Stationary Frames Applied & Computational Mathematics Seminar National University of Singapore 4PM, 20 January 2010 S16 Tutorial Room Wayne Lawton Department of Mathematics National University of Singapore

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

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

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

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 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

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

Signal Recovery the convolution property for Fourier series gives Givenwhere

Two Celebrities Recently there has been considerable interest in two deep problems that arose from very separate areas of mathematics. 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]. Kadison-Singer Problem (KSP): Does every pure state on the -subalgebraadmit 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. [KS59] R. Kadison and I. Singer, Extensions of pure states, Amer. J. Math., 81(1959), [CA05] P. G. Casazza, O. Christiansen, A. Lindner and R. Vershynin, Frames and the Feichtinger conjecture, PAMS, (4)133(2005),

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

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

Syndetic Sets and Minimal Sequences is syndetic if there exists a positive integerwith is a minimal sequence if its orbit closure Core concepts in symbolic topological dynamics [G46], [GH55] is a minimal closed shift-invariant set. [GH55] W. H. Gottschalk and G. A. Hedlund, Topological Dynamics, Amer. Math. Soc., Providence, R. I., [G46] W. H. Gottschalk, Almost periodic points with respect to transformation semigroups, Annals of Math., 47, (1946),

Symbolic Dynamics Connection the 1. following conditions are equivalent: Theorem 1.1 [LA09] For measurable is a finite union of Riesz seq. 2. There exists a syndetic set is a Riesz sequence. such that 3. There exists a nonempty set such that is a minimal sequence and is a Riesz sequence. [LA09] Minimal Sequences and the Kadison-Singer Problem, accepted BMMSS