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Sufficient conditions for irregular Gabor frames

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Abstract

Finding general and verifiable conditions which imply that Gabor systems are (resp. cannot be) Gabor frames is among the core problems in Gabor analysis. In their paper on atomic decompositions for coorbit spaces [H.G. Feichtinger and K. Gröchenig, Banach spaces related to integrable group representations, and their atomic decomposition, I, J. Funct. Anal. 86 (1989), 307–340], the authors proved that every Gabor system generated with a relatively uniformly discrete and sufficiently dense time-frequency sequence will allow series expansions for a large class of Banach spaces if the window function is nice enough. In particular, such a Gabor system is a frame for the Hilbert space of square integrable functions. However, their proof is based on abstract analysis and does not give direct information on how to determine the density in the sense of directly applicable estimates. It is the goal of this paper to present a constructive version of the proof and to provide quantitative results. Specifically, we give a criterion for the general case and explicit density for some cases. We also study the existence of Gabor frames and show that there is some smooth window function such that the corresponding Gabor system is incomplete for arbitrary time-frequency lattices.

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Authors and Affiliations

  1. Department of Mathematics, University of Vienna, Nordbergstrasse 15, A-1090, Vienna, Austria

    Hans G. Feichtinger & Wenchang Sun

  2. Department of Mathematics and LPMC, Nankai University, Tianjin, 300071, China

    Wenchang Sun

Authors
  1. Hans G. Feichtinger
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  2. Wenchang Sun
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Corresponding author

Correspondence to Hans G. Feichtinger.

Additional information

Communicated by C.A. Micchelli

Mathematics subject classifications (2000)

42C15, 42C40, 65T60

Wenchang Sun: The second author was supported by the K.C. Wong Education Foundation, the National Natural Science Foundation of China (10171050 and 10201014), and the Research Fund for the Doctoral Program of Higher Education. He thanks NuHAG at the Department of Mathematics, University of Vienna for local hospitality.

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Feichtinger, H.G., Sun, W. Sufficient conditions for irregular Gabor frames. Adv Comput Math 26, 403–430 (2007). https://doi.org/10.1007/s10444-004-7210-6

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  • DOI: https://doi.org/10.1007/s10444-004-7210-6

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