Gabor frames for rational functions

Abstract

We study the frame properties of the Gabor systems G(g;α,β):={e2πiβmxg(x−αn)}m,n∈Z.

In particular, we prove that for Herglotz windows g such systems always form a frame for L2(R) if α,β>0, αβ≤1. For general rational windows g∈L2(R) we prove that G(g;α,β) is a frame for L2(R) if 0<α,β, αβ<1, αβ∉Q and g^(ξ)≠0, 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 L2(R).