TY - JOUR

T1 - Density of Gabor Frames

A1 - Christensen,Ole

A1 - Heil,Christopher

A1 - Deng,Baiqiao

AU - Christensen,Ole

AU - Heil,Christopher

AU - Deng,Baiqiao

PB - Academic Press

PY - 1999/11

Y1 - 1999/11

N2 - A Gabor system is a set of time-frequency shifts S(g, Lambda) = {e(2 pi ibx) g(x - a)}((a,b)is an element of Lambda) of a function g is an element of L-2(R-d). We prove that if a finite union of Gabor systems boolean (ORk=1S)-S-r(g(k), Lambda(k)) forms a frame for L-2(R-d) then the lower and upper Beurling densities of Lambda = boolean ORk=1r Lambda(k) satisfy D- (Lambda) greater than or equal to 1 and D+(Lambda) < infinity. This extends recent work of Ramanathan and Steger. Additionally, we prove the conjecture that no collection boolean ORk=1r {g(k)(x - a)}(a is an element of Gamma k) of pure translates can form a frame for L-2 (R-d). (C) 1999 Academic Press.

AB - A Gabor system is a set of time-frequency shifts S(g, Lambda) = {e(2 pi ibx) g(x - a)}((a,b)is an element of Lambda) of a function g is an element of L-2(R-d). We prove that if a finite union of Gabor systems boolean (ORk=1S)-S-r(g(k), Lambda(k)) forms a frame for L-2(R-d) then the lower and upper Beurling densities of Lambda = boolean ORk=1r Lambda(k) satisfy D- (Lambda) greater than or equal to 1 and D+(Lambda) < infinity. This extends recent work of Ramanathan and Steger. Additionally, we prove the conjecture that no collection boolean ORk=1r {g(k)(x - a)}(a is an element of Gamma k) of pure translates can form a frame for L-2 (R-d). (C) 1999 Academic Press.

KW - Beurling density

KW - frame

KW - frame of translates

KW - Gabor frame

KW - Riesz basis

U2 - 10.1006/acha.1999.0271

DO - 10.1006/acha.1999.0271

JO - Applied and Computational Harmonic Analysis

JF - Applied and Computational Harmonic Analysis

SN - 1063-5203

IS - 3

VL - 7

SP - 292

EP - 304

ER -