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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On weakly almost periodic measures

Authors: Daniel Lenz and Nicolae Strungaru
Journal: Trans. Amer. Math. Soc. 371 (2019), 6843-6881
MSC (2010): Primary 11K70, 43A05, 54H20; Secondary 43A60
Published electronically: February 21, 2019
MathSciNet review: 3939563
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Abstract: We study the diffraction and dynamical properties of translation bounded weakly almost periodic measures. We prove that the dynamical hull of a weakly almost periodic measure is a weakly almost periodic dynamical system with unique minimal component given by the hull of the strongly almost periodic component of the measure. In particular the hull is minimal if and only if the measure is strongly almost periodic and the hull is always measurably conjugate to a torus and has pure point spectrum with continuous eigenfunctions. As an application we show the stability of the class of weighted Dirac combs with Meyer set or FLC support and deduce that such measures have either trivial or large pure point, respectively, continuous spectrum. We complement these results by investigating the Eberlein convolution of two weakly almost periodic measures. Here, we show that it is unique and a strongly almost periodic measure. We conclude by studying the Fourier-Bohr coefficients of weakly almost periodic measures.

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Additional Information

Daniel Lenz
Affiliation: Mathematisches Institut, Friedrich Schiller Universität Jena, 07743 Jena, Germany

Nicolae Strungaru
Affiliation: Department of Mathematical Sciences, MacEwan University, 10700 – 104 Avenue, Edmonton, Alberta, T5J 4S2 Canada — and — Department of Mathematics, Trent University, Peterborough, Ontario, Canada — and — Institute of Mathematics “Simon Stoilow”, Bucharest, Romania

Received by editor(s): February 21, 2017
Received by editor(s) in revised form: September 14, 2017, and September 28, 2017
Published electronically: February 21, 2019
Additional Notes: The second author was supported by NSERC, under grant 03762-2014, and would like to thank NSERC for their support.
Dedicated: Dedicated to Robert V. Moody on the occasion of his $75^{th}$ birthday.
Article copyright: © Copyright 2019 American Mathematical Society