Hyperbolic secants yield Gabor frames

Abstract

We show that $(g_2,a,b)$ is a Gabor frame when $a>0,b>0,ab<1$ and $g_2(t)=(1/2\pi \gamma {)}^{1/2}(\cosh \pi \gamma t{)}^{-1}$ is a hyperbolic secant with scaling parameter $\gamma >0$. This is accomplished by expressing the Zak transform of $g_2$ in terms of the Zak transform of the Gaussian $g_1(t)=(2\gamma {)}^{1/4}\exp (-\pi \gamma t^2)$, 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.