Linear independence of time-frequency translates of functions with faster than exponential decay

TL;DR

This paper proves the linear independence of time-frequency translates for functions with decay rates faster than exponential, extending known results to broader classes of functions with specific decay properties.

Contribution

It establishes linear independence for functions with super-exponential decay, including new cases with additional restrictions, advancing understanding in time-frequency analysis.

Findings

Proves linear independence for functions with decay $ o 0$ faster than $e^{cx ext{log} x}$.

Extends results to functions with decay faster than exponential $e^{cx}$ under certain conditions.

Provides theoretical foundation for the uniqueness of time-frequency representations.

Abstract

We establish the linear independence of time-frequency translates for functions $f$ having one sided decay $\lim_{x\to \infty} |f(x)| e^{cx \log x} = 0$ for all $c>0$. We also prove such results for functions with faster than exponential decay, i.e., $\lim_{x\to \infty} |f(x)| e^{cx} = 0$ for all $c>0$, under some additional restrictions.