Irregular sampling for hyperbolic secant type functions

Abstract

We study Gabor frames in the case when the window function is of hyperbolic secant type, i.e., $g(x)=(e^{ax}+e^{-bx}{)}^{-1}$, $\mathrm{Re}a,\mathrm{Re}b>0$. A criterion for half-irregular sampling is obtained: for a separated $\hat{I}›\subset \mathbb{R}$ the Gabor system $\mathcal{G}(g,\hat{I}›\times \hat{I}\pm \mathbb{Z})$ is a frame in $L^2(\mathbb{R})$ if and only if $D^{-}(\hat{I}›)>\hat{I}\pm$ where $D^{-}(\hat{I}›)$ is the usual (Beurling) lower density of $\hat{I}›$. This extends a result by Gröchenig, Romero, and Stöckler which applies to the case of a standard hyperbolic secant. Also, a full description of complete interpolating sequences for the shift-invariant space generated by $g$ is given.

14 pages