Riesz bases of exponentials on multiband spectra

TL;DR

This paper constructs Riesz bases of exponential functions for multiband spectra using quasicrystals, under specific length conditions of the intervals, advancing spectral theory and basis construction in harmonic analysis.

Contribution

It introduces a novel method using quasicrystals to build Riesz bases for multiband spectra with intervals satisfying certain length conditions.

Findings

Successfully constructs Riesz bases for specified multiband spectra.

Establishes conditions on interval lengths for basis construction.

Provides a new approach to spectral analysis using quasicrystals.

Abstract

Let $S$ be the union of finitely many disjoint intervals on the real line. Suppose that there are two real numbers $\alpha, \beta$ such that the length of each interval belongs to $Z \alpha + Z \beta$. We use quasicrystals to construct a discrete set of real frequencies such that the corresponding system of exponentials is a Riesz basis in the space $L^2(S)$.