The Periodic Decomposition Problem

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