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Diffraction as a unitary representation and the orthogonality of measures with respect to the reflected Eberlein convolution

Daniel Lenz, Nicolae Strungaru

Abstract

We discuss how the diffraction theory of a single translation bounded measure or a family of such measures can be understood within the framework of unitary group representations. This allows us to prove an orthogonality feature of measures whose diffractions are mutually singular. We apply this to study dynamical systems, the refined Eberlein decomposition and validity of a Bombieri--Taylor type result in a rather general context. Along the way we also use our approach to (re)prove various characterisations of pure point diffraction.

Diffraction as a unitary representation and the orthogonality of measures with respect to the reflected Eberlein convolution

Abstract

We discuss how the diffraction theory of a single translation bounded measure or a family of such measures can be understood within the framework of unitary group representations. This allows us to prove an orthogonality feature of measures whose diffractions are mutually singular. We apply this to study dynamical systems, the refined Eberlein decomposition and validity of a Bombieri--Taylor type result in a rather general context. Along the way we also use our approach to (re)prove various characterisations of pure point diffraction.
Paper Structure (12 sections, 42 theorems, 164 equations)

This paper contains 12 sections, 42 theorems, 164 equations.

Table of Contents

  1. The setting
  2. The reflected Eberlein convolution
  3. Existence of the reflected Eberlein convolution
  4. Construction of a unitary representation
  5. Diffraction as spectral measure
  6. An orthogonality result
  7. Application: The point part of the diffraction and Bombieri--Taylor type results
  8. Application: Existence of the refined Eberlein decomposition of an arbitrary measure
  9. Application: Orthogonality of dynamical systems
  10. Existence of the reflected Eberlein convolution for Besicovitch almost periodic measures
  11. Some background on translation bounded measures
  12. Universal van Hove nets

Key Result

Proposition 1.1

Let $\gamma$ be a positive definite measure on $G$. Then, holds for all $t\in G$ and $\varphi \in C_{\mathsf{c}} (G)$. ∎

Theorems & Definitions (96)

  • Proposition 1.1
  • Remark 1.2: Second countable LCAG
  • Definition 2.1: Reflected Eberlein convolution of measures
  • Lemma 2.2
  • proof
  • Definition 2.3: Autocorrelation and diffraction of $\mu$
  • Definition 2.4: Fourier--Bohr coefficient
  • Proposition 2.5
  • Definition 2.6: The mean $M_{\mathcal{A}}$
  • Definition 2.7: Reflected Eberlein convolution of functions
  • ...and 86 more