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Theory and Applications of Difference Equations and Discrete Dynamical Systems pp 143–169Cite as

The Periodic Decomposition Problem

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Part of the book series: Springer Proceedings in Mathematics & Statistics ((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.

Dedicated to Imre Z. Ruzsa on the occasion of his 60th birthday.

Bálint Farkas was supported in part by the Hungarian National Foundation for Scientific Research, Project #K-100461, Szilárd Gy Révész was supported in part by the Hungarian National Foundation for Scientific Research, Project #’s K-81658, K-100461, NK-104183, K-109789.

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Authors and Affiliations

  1. Bergische Universität Wuppertal, Wuppertal, Germany

    Bálint Farkas

  2. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Kuwait University, Kuwait, Kuwait

    Szilárd Gy. Révész

Authors
  1. Bálint Farkas
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  2. Szilárd Gy. Révész
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Correspondence to Szilárd Gy. Révész .

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Editors and Affiliations

  1. Sultan Qaboos University Dept. of Math. and Statistics, Muscat, Oman

    Ziyad AlSharawi

  2. University of Arizona Dept. of Mathematics, Tucson, Arizona, USA

    Jim M. Cushing

  3. Trinity University Dept. of Mathematics, San Antonio, Texas, USA

    Saber Elaydi

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Farkas, B., Révész, S.G. (2014). The Periodic Decomposition Problem. In: AlSharawi, Z., Cushing, J., Elaydi, S. (eds) Theory and Applications of Difference Equations and Discrete Dynamical Systems. Springer Proceedings in Mathematics & Statistics, vol 102. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44140-4_8

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