A New Strategy for Feichtinger’s Conjecture for Stationary Frames

Applied & Computational Mathematics SeminarNational University of Singapore

4PM, 20 January 2010     S16 Tutorial Room

Wayne LawtonDepartment of Mathematics

National University of Singaporematwml@nus.edu.sg

http://www.math.nus.edu.sg/~matwmlhttp://arxiv.org/find/math/1/au:+Lawton_W/0/1/0/all/0/1

Trigonometric Polynomials

is specified set of (integer) frequenciesZ

Λk

ktki

k CcectP :Pol 2)(

These polynomials describe functions RC having period 1.

Physical Modelskcamplitude, freq. = k component

)(tPsignal 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

Riesz-Pairs

Z

.Pol,|P(t)||P(t)|]1,0[

2

S

2 Pdtdt

is a Riesz basis, this means that there exists

such that

ket tkiS :)( 2

Definition

),( S

is measurable,If

then

and

is a Riesz-pair if

]1,0[S

0

Definition ),( S will denote the lub }0{ that satisfy the inequality above, thus RP. a is ),(0 S

Examples)],1,0([ RP for every Z

),( ZS NRP if 1)( meas S),( S NRP if the upper Beurling density

)( meas|),(|maxlim)( 1 SkaaDRakk

)],,([ ba RP if the separation,)(||min)( 1

2121

ab

),( nmZS NRP if 0m and S is nowhere dense.

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.

[MV74]

W. Lawton, Minimal Sequences and the Kadison-Singer Problem, accepted BMMSS

[LA09]

),(,0)( meas, SSS[BT87] .0|),(|mininflim)( 2

1

kkdRakk

RP and asymptoticdensity

),( S[LA09] NRP if is a Bohr minimal sequence.

Fat Cantor SetsSmith–Volterra–Cantor set (SVC) or the fat Cantor set is an example of a set of points on the real line R that is nowhere dense (in particular it contains no intervals), yet has positive measure. The Smith–Volterra–Cantor set is named after themathematicians Henry Smith, Vito Volterra and Georg 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].

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                                 

Applications

known set of possible non-zero frequency components

Robust Signal Recovery

RPSignal can be robustly recovered iff

S set over which the signal is measured

),( S

Beam Nulling

known set of transmitter locations

S set of locations where beam should be undetectable

Beam can be nulled iff ),( S NRP

Signal Recovery,1)(meas0 S,,)( 2 ZjdtetPd

St

tjij

the convolution property for Fourier series gives

Given

MMGdMGccMd TT ,1

where

k

kSZk

kSj ckjckjd )(ˆ)(ˆ

kjST kjGddM ,)(ˆ and

SsS

|||||||||||||||||| 22

122

22

122 ddGc

Two CelebritiesRecently 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 unsolvedsince it was formally stated in the literature in 2005 [CA05].

Kadison-Singer Problem (KSP): Does every pure state on the

C -subalgebra )(Z admit a unique extension to ?))(( 2 ZB arose in the area of operator algebras and has remainedunsolved since 1959 [KS59].

Feichtinger’s Conjecture (FC): Every bounded frame canbe written as a finite union of Riesz sequences.

[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, Framesand the Feichtinger conjecture, PAMS, (4)133(2005), 1025-1033.

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), 2032-2039.

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. 299-355.

Feichtinger’s Conjecture for Stationary Frames

Feichtinger’s Conjecture for Exponentials (FCE):

]1,0[S

is equivalent to the following special case of FC:

For every measurable set

mZ 1where ),( iS are RP.

[BT91] Theorem 4.1 Feichtingers conjecture holds if

.|||)(ˆ|),1,0( 2

kkZk

S

with 0)( meas S

[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 integerZ n with

.,...,2,1 Zn

Z1,0 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., 1955.

[G46] W. H. Gottschalk, Almost periodic points with respect to transformation semigroups, Annals of Math., 47, (1946), 762-766.

Symbolic Dynamics Connectionthe

Zket tkiS :)( 2

Z1.

following conditions are equivalent:Theorem 1.1 [LA09] For measurable TS

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 Z such that

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

ket tkiS :)( 2

ket tkiS :)( 2