On Eigenmeasures Under Fourier Transform

Abstract

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 $$\mathbb {R}\hspace{0.5pt}^d$$ R d . In particular, we classify all periodic eigenmeasures on $$\mathbb {R}\hspace{0.5pt}$$ R , which gives an interesting connection with the discrete Fourier transform and its eigenvectors, as well as all eigenmeasures on $$\mathbb {R}\hspace{0.5pt}$$ 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.