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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Invariant decomposition of functions with respect to commuting invertible transformations


Authors: Bálint Farkas, Viktor Harangi, Tamás Keleti and Szilárd György Révész
Journal: Proc. Amer. Math. Soc. 136 (2008), 1325-1336
MSC (2000): Primary 39A10; Secondary 39B52, 39B72.
Published electronically: December 6, 2007
MathSciNet review: 2367106
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Abstract: Consider $ a_1,\dots,a_n\in\mathbb{R}$ arbitrary elements. We characterize those functions $ f:\mathbb{R}\to\mathbb{R}$ that decompose into the sum of $ a_j$-periodic functions, i.e., $ f=f_1+\cdots+f_n$ with $ \Delta_{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 $ B_1\cup B_2\cup\cdots \cup B_N=\{a_1,\dots,a_n\}$ with $ B_j$ consisting of commensurable elements with least common multiples $ b_j$ one has $ \Delta_{{b_1}}\dots \Delta_{{b_N}}f=0$.

Actually, we prove a more general result for periodic decompositions of functions $ f:\mathcal{A}\to\mathbb{R}$ defined on an Abelian group $ \mathcal{A}$; in fact, we even consider invariant decompositions of functions $ f:A\to\mathbb{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.


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Additional Information

Bálint Farkas
Affiliation: Technische Universität Darmstadt, Fachbereich Mathematik, AG4, Schloßgartenstraße 7, D-64289, Darmstadt, Germany
Email: farkas@mathematik.tu-darmstadt.de

Viktor Harangi
Affiliation: Department of Analysis, Eötvös Loránd University, Pázmány Péter sétány 1/c, H-1117 Budapest, Hungary
Email: bizkit@cs.elte.hu

Tamás Keleti
Affiliation: Department of Analysis, Eötvös Loránd University, Pázmány Péter sétány 1/c, H-1117 Budapest, Hungary
Email: elek@cs.elte.hu

Szilárd György Révész
Affiliation: A. Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, P.O.B. 127, 1364 Hungary
Email: revesz@renyi.hu

DOI: http://dx.doi.org/10.1090/S0002-9939-07-09267-2
Keywords: Periodic functions, periodic decomposition, difference equation, commuting transformations, transformation invariant functions, difference operator, shift operator, decomposition property, Abelian groups, integer-valued functions
Received by editor(s): February 2, 2007
Published electronically: December 6, 2007
Additional Notes: Supported in the framework of the Hungarian-Spanish Scientific and Technological Governmental Cooperation, Project # E-38/04 and in the framework of the Hungarian-French Scientific and Technological Governmental Cooperation, Project # F-10/04.
The third author was supported by Hungarian Scientific Foundation grants no. F 43620 and T 49786.
This work was accomplished during the fourth author’s stay in Paris under his Marie Curie fellowship, contract # MEIF-CT-2005-022927.
Communicated by: Andreas Seeger
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.