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Abstract

Let $a_1, \ldots, a_k$ be arbitrary nonzero real numbers. An $(a_1, \ldots, a_k)$-decomposition of a function $f:\mathbb{R} \to \mathbb{R}$ is a sum $f_1 + \cdots + f_k = f$ where $f_i: \mathbb{R} \to \mathbb{R}$ is an $a_i$-periodic function. Such a decomposition is not unique because there are several solutions of the equation $h_1 + \cdots + h_k = 0$ with $h_i : \mathbb{R} \to \mathbb{R}$ $a_i$-periodic. We will give solutions of this equation with a certain simple structure (trivial solutions) and study whether there exist other solutions or not. If not, we say that the $(a_1, \ldots, a_k)$-decomposition is essentially unique. We characterize those periods for which essential uniqueness holds.