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On eigenmeasures under Fourier transform

Michael Baake, Timo Spindeler, Nicolae Strungaru

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 $\RR^d$. In particular, we classify all periodic eigenmeasures on $\RR$, which gives an interesting connection with the discrete Fourier transform and its eigenvectors, 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. Finally, we construct a large class of eigenmeasures with locally finite support that is not uniformly discrete and has large gaps around $0$.

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

Paper Structure

This paper contains 9 sections, 19 theorems, 99 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Eigenvalues and structure of solutions
  4. Lattice-periodic eigenmeasures
  5. Connection with discrete Fourier transform
  6. Shadows of product measures
  7. Eigenmeasures with uniformly discrete support
  8. Eigenmeasures with locally finite support
  9. Outlook

Key Result

Lemma \oldthetheorem

If $\mu\ne 0$ is a tempered, pure point measure on $\mathbb{R}\space^d$ with uniformly discrete support, the following properties are equivalent.

Theorems & Definitions (47)

  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • proof
  • Remark \oldthetheorem
  • ...and 37 more