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

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.

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