On the uniqueness of periodic decomposition

Viktor Harangi ¹

¹ Department of Analysis Eötvös Loránd University Pázmány Péter sétány 1//c H-1117 Budapest, Hungary

Fundamenta Mathematicae, Tome 211 (2011) no. 3, pp. 225-244

Résumé

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.

Détail

DOI : 10.4064/fm211-3-2

Keywords: ldots arbitrary nonzero real numbers ldots decomposition function mathbb mathbb sum cdots where mathbb mathbb i periodic function decomposition unique because there several solutions equation cdots mathbb mathbb i periodic solutions equation certain simple structure trivial solutions study whether there exist other solutions say ldots decomposition essentially unique characterize those periods which essential uniqueness holds