Gabor frames for rational functions

We study the frame properties of the Gabor systems $$\mathfrak{G}(g;\alpha,\beta):=\{e^{2\pi i \beta m x}g(x-\alpha n)\}_{m,n\in\mathbb{Z}}.$$ In particular, we prove that for Herglotz windows $g$ such systems always form a frame for $L^2(\mathbb{R})$ if $\alpha,\beta>0$, $\alpha\beta\leq1$. For general rational windows $g\in L^2(\mathbb{R})$ we prove that $\mathfrak{G}(g;\alpha,\beta)$ is a frame for $L^2(\mathbb{R})$ if $0<\alpha,\beta$, $\alpha\beta<1$, $\alpha\beta\not\in\mathbb{Q}$ and $\hat{g}(\xi)\neq0$, $\xi>0$, thus confirming Daubechies conjecture for this class of functions. We also discuss some related questions, in particular sampling in shift-invariant subspaces of $L^2(\mathbb{R})$.
Comment: 32 pages