Hyperbolic secants yield Gabor frames

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.