Invariant Decomposition of Functions with Respect to Commuting Invertible Transformations

Consider a1, . . . , an ∈ R arbitrary elements. We characterize those functions f : R → R that decompose into the sum of aj-periodic functions, i.e., f = f1+· · ·+fn with ∆aj f(x) := f(x+aj)−f(x) = 0. We show that f has such a decomposition if and only if for all partitions B1∪B2∪· · ·∪BN = {a1, . . . , an} with Bj consisting of commensurable elements with least common multiples bj one has ∆b1 . . .∆bN f = 0. Actually, we prove a more general result for periodic decompositions of functions f : A → R defined on an Abelian group A; in fact, we even consider invariant decompositions of functions f : A → 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.