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

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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 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.

Keywords

Almost periodic functions Besicovitch almost periodic functions Bochner’s theorem Exponential sums Fourier series 

Mathematics Subject Classification

42A75 42A16 42B05 46E30 30B50 

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of AlicanteAlicanteSpain

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