The periodic decomposition problem. (English) Zbl 1318.39003

Farkas, Bálint; Révész, Szilárd Gy.

AlSharawi, Ziyad (ed.) et al., Theory and applications of difference equations and discrete dynamical systems. ICDEA, Muscat, Oman, May 26–30, 2013. Berlin: Springer (ISBN 978-3-662-44139-8/hbk; 978-3-662-44140-4/ebook). Springer Proceedings in Mathematics & Statistics 102, 143-169 (2014).

Summary: If a function \(f:\mathbb {R}\to \mathbb {R}\) 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 \(n\)th-order difference equation \(\Delta _{\alpha _1}\cdots {\Delta} _{\alpha _n} f=0\). 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 recent results of B. Farkas [“A Bohl-Bohr-Kadets type theorem characterizing Banach spaces not containing \(c_0\)”, arXiv:1301.6250; “A note on the periodic decomposition problem for semigroups”, arXiv:1401.1226] 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.
For the entire collection see [Zbl 1297.39001].

MSC:

39A10	Additive difference equations
39A23	Periodic solutions of difference equations
39A24	Almost periodic solutions of difference equations

Keywords:

periodic functions; periodic decomposition; difference equation; almost periodic and mean periodic functions; transformation invariant functions; functions with values in a group; operator semigroups