Characterization and perturbation of Gabor frame sequences with rational parameters

M. Bownik, Ole Christensen

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    Abstract

    Let A c L-2(R) be at most countable, and E N. We characterize various frame-properties for Gabor systems of the form G(l. p/q . A) = {e(2 pi imx) g (x-np/q) : m, n epsilon Z, g epsilon A} in terms of the corresponding frame properties for the row vectors in the Zibulski-Zeevi matrix. This extends work by [Ron and Shen, Weyl-Heisenherg systenis and Riesz bases in L-2(R-d). Duke Math. J. 89 (1997) 237-282]. who considered the case where A is finite. As a consequence of the results, we obtain results concerning stability of Gabor frames under perturbation of the generators. We also introduce the concept of rigid frame sequences, which have the property that all Sufficiently small perturbations with a lower frame bound above some threshold value, automatically generate the same closed linear span. Finally, we characterize rigid Gabor frame sequences in terms of their Zibulski-Zeevi matrix.
    Original languageEnglish
    JournalJournal of Approximation Theory
    Volume147
    Issue number1
    Pages (from-to)67-80
    ISSN0021-9045
    DOIs
    Publication statusPublished - 2007

    Keywords

    • Zibulski-Zeevi transform
    • Weyl-Heisenberg frames
    • Gabor frames

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