Gabor frames for rational functions

Abstract

We study the frame properties of the Gabor systems $$\mathfrak{G}(g;α,β):=\{e^{2πi βm x}g(x-αn)\}_{m,n\in\mathbb{Z}}.$$ In particular, we prove that for Herglotz windows $g$ such systems always form a frame for $L^2(\mathbb{R})$ if $α,β>0$, $αβ\leq1$. For general rational windows $g\in L^2(\mathbb{R})$ we prove that $\mathfrak{G}(g;α,β)$ is a frame for $L^2(\mathbb{R})$ if $0<α,β$, $αβ<1$, $αβ\not\in\mathbb{Q}$ and $\hat{g}(ξ)\neq0$, $ξ>0$, thus confirming Daubechies conjecture for this class of functions. We also discuss some related questions, in particular sampling in shift-invariant subspaces of $L^2(\mathbb{R})$.