Hyperbolic secants yield Gabor frames

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

We show that $(g_2,a,b)$ is a Gabor frame when $a>0, b>0, ab<1$ and $g_2(t)=(\frac{1}{2}\pi \gamma)^{\frac{1}{2}} (\cosh \pi \gamma t)^{-1}$ 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 $g_1(t)=(2\gamma)^{\frac{1}{4}} \exp (-\pi \gamma t^2)$, 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.

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