Invariant decomposition of functions with respect to commuting invertible transformations

Consider $a_1,\dots,a_n\in\mathbb{R}$ arbitrary elements. We characterize those functions $f:\mathbb{R}\to\mathbb{R}$ that decompose into the sum of $a_j$-periodic functions, i.e., $f=f_1+\cdots+f_n$ with $\Delta_{a_j}f(x):=f(x+a_j)-f(x)=0$. We show that $f$ has such a decomposition if and only if for all partitions $B_1\cup B_2\cup\cdots \cup B_N=\{a_1,\dots,a_n\}$ with $B_j$ consisting of commensurable elements with least common multiples $b_j$ one has $\Delta_{{b_1}}\dots \Delta_{{b_N}}f=0$.

Actually, we prove a more general result for periodic decompositions of functions $f:\mathcal{A}\to\mathbb{R}$ defined on an Abelian group $\mathcal{A}$; in fact, we even consider invariant decompositions of functions $f:A\to\mathbb{R}$ with respect to commuting, invertible self-mappings of some abstract set $A$.

We also extend our results to functions between torsion free Abelian groups. As a corollary we also obtain that on a torsion free Abelian group the existence of a real-valued periodic decomposition of an integer-valued function implies the existence of an integer-valued periodic decomposition with the same periods.