$\ell ^2$-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 holds 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 nontrivial square summable function, with support in $A,$ whose partial sums of Fourier series are uniformly bounded in the uniform norm. This answers a question posed by Guido Weiss.