Hyperbolic secants yield Gabor frames

We show that (g(2), a, b) is a Gabor frame when a > 0, b > 0, ab < 1, and g(2)(t) = 1/2 pi gamma)(1/2) (cosh pigammat)(-1) is a hyperbolic secant with scaling parameter gamma > 0. This i s accomplished by expressing the Zak transform of g(2) in terms of the Zak transform of the Gaussian g(1)(t) = (2gamma)(1/4)exp(-pigammat(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(1) and g(2) are the same at critical density a = b = 1. Also, we display the "singular" dual function corresponding to the hyperbolic secant at critical density. (C) 2002 Elsevier Science (USA).