Invariant decomposition of functions with respect to commuting invertible transformations

Details Invariant decomposition of functions with respect to commuting invertible transformations Invariant decomposition of functions with respect to commuting invertible transformations

2007-03-13

arxiv.org/abs/math/0703382v2

Mathematics

Classical Analysis and ODEs

Scientific paper

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.

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