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., $g\left(x\right)={(e^{ax}+e^{-bx})}^{-1}$, $\mathrm{Re}a,\mathrm{Re}b>0$. A criterion for half-regular sampling is obtained: for a separated $\Lambda \subset R$ the Gabor system $G(g,\Lambda \times \alpha Z)$ is a frame in $L^2\left(R\right)$ if and only if $D^{-}\left(\Lambda \right)>\alpha$ where $D^{-}\left(\Lambda \right)$ is the usual (Beurling) lower density of Λ. 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.