Gabor Frames on Local Fields of Positive Characteristic

TL;DR

This paper establishes simple, verifiable conditions based on Fourier transforms that guarantee Gabor systems form frames in L^2 spaces over local fields of positive characteristic, advancing time-frequency analysis in this setting.

Contribution

It introduces new sufficient conditions for Gabor frames on local fields of positive characteristic, expressed via Fourier transforms of the generating functions.

Findings

Conditions ensure Gabor systems are frames in L^2(K)

Conditions are simple and based on Fourier transforms

Applicable to local fields of positive characteristic

Abstract

Gabor frames have gained considerable popularity during the past decade, primarily due to their substantiated applications in diverse and widespread fields of engineering and science. Finding general and verifiable conditions which imply that the Gabor systems are Gabor frames is among the core problems in time-frequency analysis. In this paper, we give some simple and sufficient conditions that ensure a Gabor system ${M_{u(m)b}T_{u(n)a}g:m,n\in \mathbb N_{0}}$ to be a frame for L^2(K). The conditions proposed are stated in terms of the Fourier transforms of the Gabor system's generating functions.