B-Spline Approximations of the Gaussian, their Gabor Frame Properties, and Approximately Dual Frames

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We prove that Gabor systems generated by certain scaled B-splines can be considered as perturbations of the Gabor systems generated by the Gaussian, with a deviation within an arbitrary small tolerance whenever the order N of the B-spline is sufficiently large. As a consequence we show that for any choice of translation/modulation parameters (Formula presented.) with (Formula presented.), the scaled version of (Formula presented.) generates Gabor frames for N sufficiently large. Considering the Gabor frame decomposition generated by the Gaussian and a dual window, the results lead to estimates of the deviation from perfect reconstruction that arise when the Gaussian is replaced by a scaled B-spline, or when the dual window of the Gaussian is replaced by certain explicitly given and compactly supported linear combinations of the B-splines. In particular, this leads to a family of approximate dual windows of a very simple form, leading to “almost perfect reconstruction� within any desired error tolerance whenever the product ab is sufficiently small. In contrast, the known (exact) dual windows have a very complicated form. A similar analysis is sketched with the scaled B-splines replaced by certain truncations of the Gaussian. As a consequence of the approach we prove (mostly known) convergence results for the considered scaled B-splines to the Gaussian in the (Formula presented.)-spaces, as well in the time-domain as in the frequency domain.
Original languageEnglish
JournalJournal of Fourier Analysis and Applications
Pages (from-to)1-22
ISSN1069-5869
DOIs
Publication statusPublished - 2017
CitationsWeb of Science® Times Cited: No match on DOI

    Research areas

  • B-Splines, Dual frames, Frames, Gaussian

ID: 138512157