Diffraction Theory and Almost Periodic Distributions

We introduce and study the notions of translation bounded tempered distributions, and autocorrelation for a tempered distribution. We further introduce the spaces of weakly, strongly and null weakly almost periodic tempered distributions and show that for weakly almost periodic tempered distribution...

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Bibliographic Details
Published inJournal of statistical physics Vol. 164; no. 5; pp. 1183 - 1216
Main Authors Strungaru, Nicolae, Terauds, Venta
Format Journal Article
LanguageEnglish
Published New York Springer US 01.09.2016
Springer
Subjects
Mathematical and Computational Physics
Physical Chemistry
Physics
Physics and Astronomy
Quantum Physics
Statistical Physics and Dynamical Systems
Theoretical
Fourier transform
Almost periodicity
Tempered distributions
Diffraction
Autocorrelation
Translation bounded distributions
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Summary:We introduce and study the notions of translation bounded tempered distributions, and autocorrelation for a tempered distribution. We further introduce the spaces of weakly, strongly and null weakly almost periodic tempered distributions and show that for weakly almost periodic tempered distributions the Eberlein decomposition holds. For translation bounded measures all these notions coincide with the classical ones. We show that tempered distributions with measure Fourier transform are weakly almost periodic and that for this class, the Eberlein decomposition is exactly the Fourier dual of the Lebesgue decomposition, with the Fourier–Bohr coefficients specifying the pure point part of the Fourier transform. We complete the project by looking at few interesting examples.
ISSN:0022-4715
1572-9613
DOI:10.1007/s10955-016-1579-8