On the Linear Independence of Finite Wavelet Systems Generated by Schwartz Functions or Functions with certain behavior at infinity

TL;DR

This paper investigates the linear independence of finite wavelet systems generated by Schwartz functions or functions with specific decay properties, providing new conditions under which these systems are linearly independent.

Contribution

It establishes the linear independence of three-point FWS generated by Schwartz functions and extends results to FWS generated by functions with particular Fourier transform behaviors.

Findings

Three-point FWS generated by Schwartz functions are linearly independent.

FWS generated by functions with Fourier transforms decreasing at infinity are linearly independent.

Linear independence holds for functions that are linear combinations of rational and exponential functions.

Abstract

One of the motivations to state HRT conjecture on the linear independence of finite Gabor systems was the fact that there are linearly dependent Finite Wavelet Systems (FWS). Meanwhile, there are also many examples of linearly independent FWS, some of which are presented in this paper. We prove the linear independence of every three point FWS generated by a nonzero Schwartz function and with any number of points if the FWS is generated by a nonzero Schwartz function, for which the absolute value of the Fourier transform is decreasing at infinity. We also prove the linear independence of any FWS generated by a nonzero square integrable function, for which the Fourier transform has certain behavior at infinity. Such a function can be any square integrable function that is a linear complex combination of real valued rational and exponential functions.