On the Stability of Gabor Frames

In this paper, we study the stability of Gabor frames {?mb,na:m,n?Z}. We show that {?mb,na:m,n?Z} remains a frame under a small perturbation of ?,m, or n. Our results improve some results from Favier and Zalik and are applicable to many frequently used Gabor frames. In particular, we study the case for which ? is not compactly supported, and, for the particular case of the Gaussian function, we give explicit stability bounds.

[1]  Dennis Gabor,et al.  Theory of communication , 1946 .

[2]  Victor Pavlovich Maslov,et al.  Advances in Soviet mathematics , 1990 .

[3]  V. Bargmann,et al.  On the Completeness of Coherent States , 1971 .

[4]  O. Christensen.  Moment Problems and Stability Results for Frames with Applications to Irregular Sampling and Gabor Frames , 1996 .

[5]  T. Strohmer,et al.  Gabor Analysis and Algorithms: Theory and Applications , 1997 .

[6]  O. Christensen.  A Paley-Wiener theorem for frames , 1995 .

[7]  R. A. Zalik,et al.  On the Stability of Frames and Riesz Bases , 1995 .

[8]  Yurii Lyubarskii.  Frames in the Bargmann space of entire functions , 1992 .

[9]  A. Perelomov.  On the completeness of a system of coherent states , 1971, math-ph/0210005.

[10]  I. Daubechies,et al.  Frames in the Bargmann Space of Entire Functions , 1988 .

[11]  A. Grossmann,et al.  Proof of completeness of lattice states in the k q representation , 1975 .

[12]  Ingrid Daubechies,et al.  The wavelet transform, time-frequency localization and signal analysis , 1990, IEEE Trans. Inf. Theory.

[13]  K. Seip.  Density theorems for sampling and interpolation in the Bargmann-Fock space I , 1992, math/9204238.

[14]  Ingrid Daubechies,et al.  Ten Lectures on Wavelets , 1992 .

[15]  R. Young,et al.  An introduction to non-harmonic Fourier series , 2001 .

[16]  I. Daubechies,et al.  Gabor Time-Frequency Lattices and the Wexler-Raz Identity , 1994 .

[17]  O. Christensen,et al.  Perturbation of operators and applications to frame theory , 1997 .