On Eigenmeasures Under Fourier Transform

Abstract

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 mathbb {R}hspace{0.5pt}^d. In particular, we classify all periodic eigenmeasures on mathbb {R}hspace{0.5pt}, which gives an interesting connection with the discrete Fourier transform and its eigenvectors, as well as all eigenmeasures on mathbb {R}hspace{0.5pt} with uniformly discrete support. An interesting subclass of the latter emerges from the classic cut and project method for aperiodic Meyer sets. Finally, we construct a large class of eigenmeasures with locally finite support that is not uniformly discrete and has large gaps around 0.