Linear independence of time frequency translates for special configurations

TL;DR

This paper establishes linear independence of certain time-frequency translates for special point configurations, extending to $L^2( )$ functions under mild Diophantine conditions, with implications for harmonic analysis.

Contribution

It proves linear independence of time-frequency translates for four points on two parallel lines, a novel result in harmonic analysis.

Findings

Linear independence holds for four points on two parallel lines.

Extends to $L^2( )$ functions under mild Diophantine conditions.

No linear dependence exists for these configurations in the specified settings.

Abstract

We prove that for any 4 points in the plane that belong to 2 parallel lines, there is no linear dependence between the associated time-frequency translates of any nontrivial Schwartz function. If mild Diophantine properties are satisfied, we also prove linear independence in the category of $L^2(\R)$ functions.