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The Ramanujan Journal

, Volume 46, Issue 1, pp 245–267 | Cite as

Almost periodic functions in terms of Bohr’s equivalence relation

  • J. M. Sepulcre
  • T. Vidal
Article

Abstract

In this paper we introduce an equivalence relation on the classes of almost periodic functions of a real or complex variable which is used to refine Bochner’s result that characterizes these spaces of functions. In fact, with respect to the topology of uniform convergence, we prove that the limit points of the family of translates of an almost periodic function are precisely the functions which are equivalent to it, which leads us to a characterization of almost periodicity. In particular we show that any exponential sum which is equivalent to the Riemann zeta function, \(\zeta (s)\), can be uniformly approximated in \(\{s=\sigma +it:\sigma >1\}\) by certain vertical translates of \(\zeta (s)\).

Keywords

Almost periodic functions Exponential sums Riemann zeta function Bochner’s theorem Fourier series Dirichlet series 

Mathematics Subject Classification

30D20 30B50 11K60 30Axx 

Notes

Acknowledgements

The authors thank the anonymous referee for his/her valuable comments on our manuscript which led us to generalize our results.

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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of AlicanteAlicanteSpain

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