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

TL;DR

This paper introduces a new equivalence relation for Besicovitch almost periodic functions inspired by Bohr's relation, demonstrating properties like compactness of classes and density of translates in a key subspace.

Contribution

It extends Bohr's equivalence relation to Besicovitch almost periodic functions and analyzes the structure of equivalence classes within a significant subspace.

Findings

Equivalence classes in B^2 are sequentially compact.

Translates of functions are dense within their equivalence classes.

The new relation is defined via polynomial approximations.

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.