Hyperbolic secants yield Gabor frames

A.J.E.M. Janssen, T. Strohmer

Research output: Contribution to journalArticleAcademicpeer-review

41 Citations (Scopus)

Abstract

We show that (g2,a,b) is a Gabor frame when a > 0, b > 0, ab<1, and g2(t)=(12p¿)1/2(coshp¿t) -1 is a hyperbolic secant with scaling parameter ¿>0. This is accomplished by expressing the Zak transform of g2 in terms of the Zak transform of the Gaussian g1(t)=(2¿)1/4 exp(-p¿t2), together with an appropriate use of the Ron-Shen criterion for being a Gabor frame. As a side result it follows that the windows, generating tight Gabor frames, that are canonically associated to g2 and g1 are the same at critical density a=b=1. Also, we display the "singular" dual function corresponding to the hyperbolic secant at critical density.
Original languageEnglish
Pages (from-to)259-267
Number of pages9
JournalApplied and Computational Harmonic Analysis
Volume12
Issue number2
DOIs
Publication statusPublished - 2002

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