On Eigenmeasures Under Fourier Transform
Tóm tắt
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.
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