Gabor frames with rational density

TL;DR

This paper investigates the frame properties of Gabor systems with rational density, introducing a rational Gramian and demonstrating non-frame conditions for odd window functions at specific densities, with focus on the Hermite function.

Contribution

It introduces a rational analogue of the Ron-Shen Gramian and establishes non-frame conditions for Gabor systems with odd windows at certain rational densities.

Findings

Gabor systems with odd windows do not form frames when αβ = (n-1)/n.

A rational Gramian is constructed for analyzing frame properties.

Special attention is given to the Hermite function h_1(t).

Abstract

We consider the frame property of the Gabor system G(g, {\alpha}, {\beta}) = {e2{\pi}i{\beta}nt g(t - {\alpha}m) : m, n \in Z} for the case of rational oversampling, i.e. {\alpha}, {\beta} \in Q. A 'rational' analogue of the Ron-Shen Gramian is constructed, and prove that for any odd window function g the system G(g, {\alpha}, {\beta}) does not generate a frame if {\alpha}{\beta} = (n-1)/n. Special attention is paid to the first Hermite function h_1(t) = te^(-{\pi}t^2).