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.