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 \(\zeta (s)\).
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01 March 2019
The authors wish to draw the attention to a mistake which appears in the proof of Proposition 3 of the above quoted paper [4].
01 March 2019
The authors wish to draw the attention to a mistake which appears in the proof of Proposition 3 of the above quoted paper [4].
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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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The first author’s research was partially supported by Generalitat Valenciana under Project GV/2015/035.
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Sepulcre, J.M., Vidal, T. Almost periodic functions in terms of Bohr’s equivalence relation. Ramanujan J 46, 245–267 (2018). https://doi.org/10.1007/s11139-017-9950-1
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DOI: https://doi.org/10.1007/s11139-017-9950-1
Keywords
- Almost periodic functions
- Exponential sums
- Riemann zeta function
- Bochner’s theorem
- Fourier series
- Dirichlet series