Almost periodic functions in terms of Bohr’s equivalence relation. (English) Zbl 1391.30043
Ramanujan J. 46, No. 1, 245-267 (2018); correction ibid. 48, No. 3, 685-690 (2019).
Summary: 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 \(\zeta (s)\).
MSC:
| 30D20 | Entire functions of one complex variable (general theory) |
| 30B50 | Dirichlet series, exponential series and other series in one complex variable |
| 30E10 | Approximation in the complex plane |
Keywords:
entire functions; almost periodic functions; exponential sums; Riemann zeta function; Fourier series; Dirichlet seriesReferences:
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