$\ell^2$-Linear Independence for the System of Integer Translates of a Square Integrable Function

TL;DR

This paper establishes that the $ ext{l}^2$-linear independence of integer translates of a square integrable function implies its periodization function is positive almost everywhere, with implications for Fourier analysis and function support.

Contribution

It proves a new link between $ ext{l}^2$-linear independence of translates and the positivity of the periodization function for square integrable functions.

Findings

$ ext{l}^2$-linear independence implies positive periodization function

Existence of square summable functions with bounded Fourier partial sums on subsets of [0,1]

Extension of Fourier analysis results to support properties of functions

Abstract

We prove that if the system of integer translates of a square integrable function is $\ell^2$-linear independent then its periodization function is strictly positive almost everywhere. Indeed we show that the above inference is true for any square integrable function since the following statement on Fourier analysis is true: For any (Lebesgue) measurable subset A of [0,1], with positive measure, there exists a non trivial square summable function, with support in A, whose partial sums of Fourier series are uniformly bounded.