Gabor Frames and Totally Positive Functions

TL;DR

This paper proves that Gabor systems generated by totally positive functions of finite type form frames for L^2(R) if and only if the product of the lattice parameters is less than one, advancing the understanding of Gabor frame conditions.

Contribution

It provides the first positive result for a broad class of functions, confirming a longstanding conjecture and expanding the known conditions for Gabor frame generation.

Findings

Gabor systems with totally positive functions form frames if and only if αβ<1.

Introduces an uncountable class of functions generating Gabor frames.

Derives new sampling theorems and determines the correct Nyquist rate.

Abstract

Let $g$ be a totally positive function of finite type. Then the Gabor set $\{e^{2\pi i \beta l t} g(t-\alpha k), k,l \in Z \}$ is a frame for $L^2(R)$, if and only if $\alpha \beta <1$. This result is a first positive contribution to a conjecture of I.\ Daubechies from 1990. So far the complete characterization of lattice parameters $\alpha, \beta $ that generate a frame has been known for only six window functions $g$. Our main result now provides an uncountable class of functions. As a byproduct of the proof method we derive new sampling theorems in shift-invariant spaces and obtain the correct Nyquist rate.