Isospectral measures

TL;DR

This paper investigates measures in Euclidean space that admit a Fourier basis of complex exponentials for a fixed spectrum, exploring the conditions and possibilities for such measures to exist.

Contribution

It characterizes measures with a given spectrum, advancing understanding of spectral measures and their Fourier bases in Euclidean spaces.

Findings

Identifies conditions under which measures have a specified spectrum.

Provides classifications of spectral measures based on fixed spectra.

Explores the structure of measures with prescribed Fourier bases.

Abstract

In recent papers a number of authors have considered Borel probability measures $\mu$ in $\br^d$ such that the Hilbert space $L^2(\mu)$ has a Fourier basis (orthogonal) of complex exponentials. If $\mu$ satisfies this property, the set of frequencies in this set are called a spectrum for $\mu$. Here we fix a spectrum, say $\Gamma$, and we study the possibilities for measures $\mu$ having $\Gamma$ as spectrum.