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Hyperbolic secants yield Gabor frames

By A.J.E.M. Janssen and Thomas Strohmer

Abstract

We show that (g 2 ; a; b) is a Gabor frame when a > 0; b > 0; ab < 1 and g 2 (t) = ( 1 2 ) 1 2 (cosh t) 1 is a hyperbolic secant with scaling parameter > 0. This is accomplished by expressing the Zak transform of g 2 in terms of the Zak transform of the Gaussian g 1 (t) = (2) 1 4 exp( t 2 ), 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 g 2 and g 1 are the same at critical density a = b = 1. Also, we display the \singular" dual function corresponding to the hyperbolic secant at critical density. AMS Subject Classication: 42C15, 33D10, 94A12. Key words: Gabor frame, Zak transform, hyperbolic secant, theta functions.

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