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

Abstract

Based on Bohr's equivalence relation which was established for general Dirichlet series, in this paper we introduce a new equivalence relation on the space of almost periodic functions in the sense of Besicovitch, B(R, C), defined in terms of polynomial approximations. From this, we show that in an important subspace B 2 (R, C) ⊂ B(R, C), where Parseval's equality and Riesz-Fischer theorem holds, its equivalence classes are sequentially compact and the family of translates of a function belonging to this subspace is dense in its own class.

Bochner-Type Property on Spaces of Generalized Almost Periodic Functions

“…In this paper we show that the Bochner-type property, which is satisfied for all these classes of generalized almost periodic functions, can be refined in the sense that the condition of almost periodicity (in AP (R, C), S p (R, C), W p (R, C) or B p (R, C)) implies that every sequence of translates has a subsequence that converges, with respect to the topology of the corresponding space, to an equivalent function (see Theorem 14 and Corollary 16). This means that, while it is true that the proofs of the intermediate results of this paper are notably different from those of [14,15], the main results of both papers [14,15] are now extended to all these spaces. In fact, we can go further by extending these results to every existing space of generalized almost periodic functions satisfying the appropriate conditions (see Remark 17).…”

“…We next consider the same equivalence relation on the classes S Λ as that of [15,Definition 2]. Definition 2 Given an arbitrary countable set Λ = {λ 1 , λ 2 , .…”

“…By abuse of notation, we will say that G Λ is a basis for Λ. Moreover, we will say that G Λ is an integral basis for Λ when r j,k ∈ Z for any j, k. Now, the equivalence relation introduced in Definition 2 can be characterized in terms of a basis for Λ (see the proof in [15,Proposition 1]).…”

“…This is not the case for some Stepanov or Weyl functions [11]. Note that Riesz-Fischer theorem was used in [15] in order to obtain our results for these spaces (see the proof of [15,Lemma 1]).…”

“…Precisely, in the context of the Besicovitch almost periodic functions B(R, C), among which it was considered an equivalence relation in terms similar to Bohr's equivalence relation on general Dirichlet series (see [15,Definition 5]), the main result of [15] states that, given an almost periodic function in B 2 (R, C), the limit points of the set of its translates are precisely the functions which are equivalent to it. Likewise, in terms of this equivalence relation on the space AP (R, C), the paper [14] refined Bochner's result in the sense that we proved that the condition of almost periodicity (in AP (R, C)) is equivalent to that every sequence of translates has a subsequence that converges uniformly to an equivalent function.…”