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Invariant decomposition of functions with respect to commuting invertible transformations |
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| Authors: | Bá lint Farkas Viktor Harangi Tamá s Keleti Szilá rd Gyö rgy Ré vé sz |
| Affiliation: | Technische Universität Darmstadt, Fachbereich Mathematik, AG4, Schloß{}gartenstraß{}e 7, D-64289, Darmstadt, Germany ; Department of Analysis, Eötvös Loránd University, Pázmány Péter sétány 1/c, H-1117 Budapest, Hungary ; Department of Analysis, Eötvös Loránd University, Pázmány Péter sétány 1/c, H-1117 Budapest, Hungary ; A. Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, P.O.B.~127, 1364 Hungary |
| Abstract: | Consider  arbitrary elements. We characterize those functions  that decompose into the sum of  -periodic functions, i.e.,  with  . We show that  has such a decomposition if and only if for all partitions  with  consisting of commensurable elements with least common multiples  one has  . Actually, we prove a more general result for periodic decompositions of functions 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. |
| Keywords: | Periodic functions periodic decomposition difference equation commuting transformations transformation invariant functions difference operator shift operator decomposition property Abelian groups integer-valued functions |
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