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 | References | Similar Articles | Additional Information

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 defined on an Abelian group ; in fact, we even consider invariant decompositions of functions with respect to commuting, invertible self-mappings of some abstract set .

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.