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

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.

arXiv commentThis paper has been withdrawn by the author due to a crucial error in the main result of the first version. The abstract has been changed accordingly. A new, corrected version of this paper has been published elsewhere