Gabor frames for rational functions

Abstract

We study the frame properties of the Gabor systems $$\mathfrak{G}(g;\alpha ,\beta ):=\{e^{2\pi i\beta mx}g(x-\alpha n){\}}_{m,n\in \mathbb{Z}}.$$ In particular, we prove that for Herglotz windows $g$ such systems always form a frame for $L^2(\mathbb{R})$ if $\alpha ,\beta >0$, $\alpha \beta \le 1$. 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 $0<\alpha ,\beta$, $\alpha \beta <1$, $\alpha \beta \notin \mathbb{Q}$ and $\hat{g}(\xi )\ne 0$, $\xi >0$, thus confirming Daubechies conjecture for this class of functions. We also discuss some related questions, in particular sampling in shift-invariant subspaces of $L^2(\mathbb{R})$.

32 pages