Exponential Riesz bases in $L^2$ on two interval

TL;DR

This paper establishes sufficient conditions for exponential systems to form Riesz bases in $L^2$ spaces over two intervals, revealing near-necessity of these conditions and an 'extra point effect' phenomenon.

Contribution

It provides new criteria for Riesz basis formation in union of two intervals and uncovers the 'extra point effect' in such systems.

Findings

Conditions are close to necessary for Riesz bases in two-interval $L^2$ spaces.

Demonstrates the 'extra point effect' where the basis differs by one point from an interval basis.

Provides insights into the structure of exponential Riesz bases in multi-interval domains.

Abstract

We give sufficient conditions for the exponential system to be a Riesz basis in $L^2(E)$, where $E$ is a union of two intervals. We show that these conditions are close to be necessary. In addition, we demonstrate ``extra point effect'' for such systems, i.e. it may happen that the Riesz basis in $L^2(E)$ differs by one point from the Riesz basis on an interval.