Almost periodic functions in terms of Bohr’s equivalence relation

Abstract

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 + it : σ > 1} by certain vertical translates of ζ(s).

A Generalization of Bohr’s Equivalence Theorem

“…Hence j≥1 a j e <rj,x0>i e λj s is the associated Dirichlet series of an almost periodic function h(s) ∈ AP (U, C) such that h ∼ f (see [13,Lemma 3]) and, by taking Remark 1 into account, we have that w 0 = h(σ 0 + it 0 ), which shows that w 0 ∈ f k ∼f Img (f k (σ 0 + it)) .…”

“…Take h n (s) := f 2 (s + it n ), n ∈ N. By [13,Proposition 4], there exists a subsequence {h n k } k ⊂ {h n } n which converges uniformly on compact subsets to a function h(s), with h ∼ f 2 . Observe that lim k→∞ h n k (σ 0 ) = h(σ 0 ) = w 0 .…”

“….) ∈ R ♯GΛ such that b j = a j e <rj,x0>i for every j ≥ 1 (see [13,Proposition 1]). In fact, all the results of [13] which can me formulated in terms of an integral basis are also valid under Definition 2.…”

“…where the λ j 's are complex numbers and the P j (p)'s are polynomials in the parameter p. Precisely, by analogy with Bohr's theory, we established in [13] an equivalence relation ∼ on the classes S Λ consisting of exponential sums of the form j≥1 a j e λj p , a j ∈ C, λ j ∈ Λ,…”

Bohr’s equivalence relation in the space of Besicovitch almost periodic functions

“…as exponential sums, where the frequencies λ j are complex numbers and the P j (p) are polynomials in p. In this paper we are going to consider some functions which are associated with a concrete subclass of these exponential sums, where the parameter p will be changed by t in the real case. In this way, as in [11], we take the following definition.…”