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The Periodic Decomposition Problem

  • Bálint Farkas
  • Szilárd  Gy. RévészEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 102)

Abstract

If a function \(f:\mathbb {R}\rightarrow \mathbb {R}\) can be represented as the sum of \(n\) periodic functions as \(f=f_1+\cdots +f_n\) with \(f(x+\alpha _j)=f(x)\) (\(j=1,\dots ,n\)), then it also satisfies a corresponding nth order difference equation \(\Delta _{\alpha _1}\dots \Delta _{\alpha _n} f=0\). The periodic decomposition problem asks for the converse implication, which may hold or fail depending on the context (on the system of periods, on the function class in which the problem is considered, etc.). The problem has natural extensions and ramifications in various directions, and is related to several other problems in real analysis, Fourier and functional analysis. We give a survey about the available methods and results, and present a number of intriguing open problems. Most results have already appeared elsewhere, while the recent results of [7, 8] are under publication. We give only some selected proofs, including some alternative ones which have not been published, give substantial insight into the subject matter, or reveal connections to other mathematical areas. Of course this selection reflects our personal judgment. All other proofs are omitted or only sketched.

Keywords

Periodic functions Periodic decomposition Difference equation Almost periodic and mean periodic functions Transformation invariant functions Functions with values in a group Operator semigroups 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Bergische Universität WuppertalWuppertalGermany
  2. 2.Alfréd Rényi Institute of Mathematics, Hungarian Academy of SciencesKuwait UniversityKuwaitKuwait

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