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 Rd. In particular, we classify all periodic eigenmeasures on R, which gives an interesting connection with the discrete Fourier transform and its eigenvectors, as well as all eigenmeasures on R 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.