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, $B(\mathbb{R},\mathbb{C})$, defined in terms of polynomial approximations. From this, we show that in an important subspace $B^2(\mathbb{R},\mathbb{C})\subset B(\mathbb{R},\mathbb{C})$, 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.

Because of a mistake detected in one of the references, the equivalence relation which is inspired by that of Bohr is revised to adapt correctly the situation in the general case. arXiv admin note: text overlap with arXiv:1801.08035