The periodic decomposition problem

If a function $f:\mathbb{R}\to\mathbb{R}$ can be represented as the sum of $n$ periodic functions as $f=f_1+\dots+f_n$ with $f(x+α_j)=f(x)$ ($j=1,\dots,n$), then it also satisfies a corresponding $n$-order difference equation $Δ_{α_1}\dotsΔ_{α_n} f=0$. 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.

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