Riesz bases of exponentials for finite unions of intervals

Abstract

This thesis presents a proof of the existence of exponential Riesz bases for the space of square-integrable functions defined on a finite union of intervals. For this purpose, we begin with the study of basis theory in Hilbert spaces. In this context, we define Riesz bases, which can be understood as controlled deformations of orthonormal bases.

Subsequently, we particularize the results obtained for Riesz bases to the space $L^2$. Our analysis focuses on the strong connection between square-integrable functions defined on a bounded interval and entire functions of exponential type. This relation is established by the Fourier transform. In this part, we follow the work of Paley and Wiener.

Finally, we introduce the necessary framework to combine Riesz bases, which constitutes the essential tool to prove our main result. This thesis concludes with the existence proof originally developed by Gady Kozma and Nitzan Shahaf.