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.
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