  Consider a_1,a_2,...,a_n, arbitrary elements of R. We characterize those real
functions f that decompose into the sum of a_j-periodic functions, i.e.,
f=f_1+...+f_n with D_{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 to B_1, B_2,... B_N of
{a_1,a_2,...,a_n} with B_j consisting of commensurable elements with least
common multiples b_j, one has D_{b_1}... D_{b_N}f=0.
  Actually, we prove a more general result for periodic decompositions of real
functions f defined on an Abelian group A, and, in fact, we even consider
invariant decompositions of functions f defined on some abstract set A, with
respect to commuting, invertible self-mappings of the 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.
