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On the Fourier Transformability of Strongly Almost Periodic Measures

Part of: Harmonic analysis in one variable Abstract harmonic analysis

Published online by Cambridge University Press:  29 January 2019

Nicolae Strungaru
Nicolae Strungaru*
Affiliation:
Department of Mathematical Sciences, MacEwan University, 10700 – 104 Avenue, Edmonton, AB, T5J 4S2 Department of Mathematics, Trent University, Peterborough, ON Institute of Mathematics “Simon Stoilow”, Bucharest, Romania Email: strungarun@macewan.ca URL: http://academic.macewan.ca/strungarun/
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Abstract

In this paper we characterize the Fourier transformability of strongly almost periodic measures in terms of an integrability condition for their Fourier–Bohr series. We also provide a necessary and sufficient condition for a strongly almost periodic measure to be the Fourier transform of a measure. We discuss the Fourier transformability of a measure on $\mathbb{R}^{d}$ in terms of its Fourier transform as a tempered distribution. We conclude by looking at a large class of such measures coming from the cut and project formalism.

Keywords

Fourier transform of measuresalmost periodic measureFourier-Bohr seriesEberlein decomposition

MSC classification

Primary: 43A05: Measures on groups and semigroups, etc.
Secondary: 43A60: Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions 43A25: Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups 42A75: Classical almost periodic functions, mean periodic functions
Type
Article
Information
Canadian Journal of Mathematics , Volume 72 , Issue 4 , August 2020 , pp. 900 - 927
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Copyright
© Canadian Mathematical Society 2019

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