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.
History
Publication title
Journal of Statistical Physics
Volume
164
Issue
5
Pagination
1183-1216
ISSN
0022-4715
Department/School
School of Natural Sciences
Publisher
Kluwer Academic/Plenum Publ
Place of publication
233 Spring St, New York, USA, Ny, 10013
Rights statement
Copyright 2016 Springer Science+Business Media New York