Exponential spectra in $L^2(\mu)$

TL;DR

This paper investigates the structure of exponential bases and frames in $L^2()$ spaces, classifies measures that admit such bases, and constructs novel examples of singularly continuous measures with unique basis properties.

Contribution

It establishes that measures with exponential frames must be of pure type, classifies measures admitting exponential bases, and constructs the first example of a singularly continuous measure with an exponential Riesz basis but no orthonormal basis.

Findings

Exponential frames imply the measure is of pure type.

Classification of measures with exponential bases, including discrete measures and their relation to integer tiles.

Construction of a singularly continuous measure with an exponential Riesz basis but no orthonormal basis.

Abstract

Let $\mu$ be a Borel probability measure with compact support. We consider exponential type orthonormal bases, Riesz bases and frames in $L^2(\mu)$. We show that if $L^2(\mu)$ admits an exponential frame, then $\mu$ must be of pure type. We also classify various $\mu$ that admits either kind of exponential bases, in particular, the discrete measures and their connection with integer tiles. By using this and convolution, we construct a class of singularly continuous measures that has an exponential Riesz basis but no exponential orthonormal basis. It is the first of such kind of examples.