Gabor frames for rational functions

TL;DR

This paper investigates the conditions under which Gabor systems with rational functions as windows form frames in L^2(R), confirming a conjecture for a broad class of functions and exploring related sampling questions.

Contribution

It establishes new frame conditions for Gabor systems with rational and Herglotz windows, confirming Daubechies conjecture for these classes and analyzing sampling in shift-invariant spaces.

Findings

Gabor systems with Herglotz windows always form frames if αβ ≤ 1.

For rational windows, frames exist if 0<α,β, αβ<1, αβ not rational, and the Fourier transform of g is non-zero on positive reals.

Confirmed Daubechies conjecture for a broad class of rational functions.

Abstract

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 $L^2(\mathbb{R})$.