2007/03/13 by Bálint Farkas, Viktor Harangi, Farkas, Bálint +5
Mathematics · #Advanced Topology and Set Theory #Mathematical Dynamics and Fractals #advanced mathematical theories #math.CA #msc:39A10. #msc:39B52 #msc:39B72
paper · pdf · doi:10.48550/arxiv.math/0703382
arxiv created 2007/05/25 · arxiv updated 2009/12/01
Consider a1,a2,...,an, arbitrary elements of R. We characterize those real functions f that decompose into the sum of aj-periodic functions, i.e., f=f1+...+fn with Dajf(x):=f(x+aj)-f(x)=0. We show that f has such a decomposition if and only if for all partitions to B1, B2,... BN of a1,a2,...,an with Bj consisting of commensurable elements with least common multiples bj, one has Db1... DbNf=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.