**ABSTRACT**
### 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.

Download the paper as a GNU-ziped postscript file (73390 bytes).