Several classes of tempered measures are characterised that are eigenmeasures of the Fourier transform, the latter viewed as a linear operator on (generally unbounded) Radon measures on $RR^d$. In particular, we classify all periodic eigenmeasures on $RR$, which gives an interesting connection with the discrete Fourier transform, as well as all eigenmeasures on $RR$ with uniformly discrete support. An interesting subclass of the latter emerges from the classic cut and project method for aperiodic Meyer sets.
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