Density of Gabor Schauder bases

Baiqiao Deng; Christopher E. Heil

doi:10.1117/12.408600

4 December 2000 Density of Gabor Schauder bases

Baiqiao Deng, Christopher E. Heil

Proceedings Volume 4119, Wavelet Applications in Signal and Image Processing VIII; (2000) https://doi.org/10.1117/12.408600
Event: International Symposium on Optical Science and Technology, 2000, San Diego, CA, United States

Abstract

A Gabor system is a fixed set of time-frequency shifts G(g, (Lambda) ) equals [e^2(pi ib x)g(x-a)] (a,b) (epsilon) (Lambda) of a function g (epsilon) L²(R^d). We prove that if G(g, (Lambda) ) forms a Schauder basis for L²(R^d) then the upper Beurling density of (Lambda) satisfies D⁺((Lambda) ) <EQ 1. We also prove that if G(g, (Lambda) ) forms a Schauder basis for L²(R^d) and if g lies in a the modulation space M^1,1(R^d), which is a dense subset of L²(R^d), or if G(g, (Lambda) ) possesses at least a lower frame bound, then (Lambda) has uniform Beurling density D((Lambda) ) equals 1. We use related techniques to show that if g (epsilon) L¹(R^d) then no collection [ g(x-a)]_{a (epsilon} (Gamma) ) of pure translates of g can form a Schauder basis for L²(R^d). We also extend these results to the case of finitely many generating functions g_l,...,g_r.

Citation Download Citation

Baiqiao Deng and Christopher E. Heil "Density of Gabor Schauder bases", Proc. SPIE 4119, Wavelet Applications in Signal and Image Processing VIII, (4 December 2000); https://doi.org/10.1117/12.408600

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