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

2017/11/11 by J. M. Sepulcre, T. Vidal, Sepulcre, J. M. +1
Mathematics · #30B50 #30Bxx #42A16 #42A75 #42Axx #42B05 #46xx #Advanced Banach Space Theory #FOS: Mathematics #Functional Analysis (math.FA) #Functional Equations Stability Results #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1711.04122

openalex publication_date 2017/11/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

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 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.

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