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 B2(R,C)⊂B(R,C), where Parseval’s equality and the Riesz–Fischer theorem hold, its equivalence classes are sequentially compact and the family of translates of a function belonging to this subspace is dense in its own class.