Sepulcre, Juan Matias, Vidal, Tomás
Almost periodic functions in terms of Bohr’s equivalence relation
The Ramanujan Journal. 2018, 46(1): 245-267. doi:10.1007/s11139-017-9950-1
URI: http://hdl.handle.net/10045/74790
DOI: 10.1007/s11139-017-9950-1
ISSN: 1382-4090 (Print)
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, ζ(s), can be uniformly approximated in {s = σ +i t : σ > 1} by certain vertical translates of ζ(s).
Keywords:Almost periodic functions, Exponential sums, Riemann zeta function, Bochner’s theorem, Fourier series, Dirichlet series
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info:eu-repo/semantics/article