Totally Positive Functions and Gabor Frames over Rational Lattices

TL;DR

This paper proves that Gabor systems generated by totally positive functions form frames for L^2(R) over rational lattices precisely when the product of the lattice parameters is less than one.

Contribution

It establishes a necessary and sufficient condition for Gabor frames with totally positive functions over rational lattices, extending the understanding of frame conditions in time-frequency analysis.

Findings

Gabor family forms a frame if and only if αβ<1.

The result applies to any totally positive function in L^1(R).

Frame property depends solely on the product of lattice parameters.

Abstract

We show that for an arbitrary totally positive function $g\in L^1(\mathbb{R} )$ and $\alpha \beta$ rational, the Gabor family $\{e^{2\pi i \beta l t} g(t-\alpha k): k,l \in \mathbb{Z} \}$ is a frame for $L^2(\mathbb{R})$, if and only if $\alpha \beta <1$.