Exponential spectra in $L^2(μ)$

Abstract

Let $μ$ be a Borel probability measure with compact support. We consider exponential type orthonormal bases, Riesz bases and frames in $L^2(μ)$. We show that if $L^2(μ)$ admits an exponential frame, then $μ$ must be of pure type. We also classify various $μ$ 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.