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<p>We discuss how the diffraction theory of a single translation bounded measure or a family of such measures can be placed 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.</p>

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Diffraction as a Unitary Representation and the Orthogonality of Measures with Respect to the Reflected Eberlein Convolution

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  • 发表日期：

    2026年7月21日

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  • Daniel Lenz,
  • Nicolae Strungaru

摘要

We discuss how the diffraction theory of a single translation bounded measure or a family of such measures can be placed 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.