Abstract
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.
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Notes
See [7, Def. 15.1] for the definition of Eberlein convolution of weakly almost periodic functions.
Recall that \(\Lambda \) is said to have finite local complexity if the Minkowski difference \(\Lambda -\Lambda := \{ x-y | x,y \in \Lambda \}\) has finite intersection with all compact sets.
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Acknowledgments
The authors are grateful to Michael Baake for stimulating our interest in these questions, as well as for numerous discussions and comments which improved the quality of this manuscript. Part of the work was done while NS visited the University of Bielefeld and NS would like to thank the University for the hospitality. The work was partially supported by the German Research Foundation (DFG), within the CRC 701 and partially supported by NSERC with a research grant number 2014-03762 and the authors are grateful for the support. We would also like to thank the anonymous referee for a careful reading of the manuscript and for making suggestions which improved the quality of the paper.
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Strungaru, N., Terauds, V. Diffraction Theory and Almost Periodic Distributions. J Stat Phys 164, 1183–1216 (2016). https://doi.org/10.1007/s10955-016-1579-8
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DOI: https://doi.org/10.1007/s10955-016-1579-8