The Periodic Decomposition Problem

Abstract

Dedicated to Imre Z. Ruzsa on the occasion of his 60 th birthday.Abstract If a function f : R → R can be represented as the sum of n periodic functions as f = f 1 + · · · + f n with f (x + α j ) = f (x) (j = 1, . . . , n), then it also satisfies a corresponding n-order difference equation ∆ α1 . . . ∆ αn 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.

Sums and products of periodic functions

“…More generally, if ∆ T denotes the linear operator defined by ∆ T f (x) = f (x + T ) − f (x), Mortola and R. Peirone [4] have shown that if (Wierdl [5] proved before one implication). This result is extended in [1] and [2] in more general settings. It implies in particular that if, for some integer n ≥ 1, f (x) = x n for all x ∈ R, then f can be expressed as the sum of n + 1 periodic functions, but cannot be written as the sum of n periodic functions.…”

“…We will see that it is indeed possible whenever f (x) = x for all x. However, M. Laczkovich and S. G. Révész obtained in [3] that in this case, f 1 and f 2 are not measurable (see also [2]). On the other hand, if, for some n ≥ 2, f (x) = x n for all x ∈ R, the question has a negative answer.…”