Article

Aequationes mathematicae

, Volume 73, Issue 3, pp 233-248

First online:

Decomposition as the sum of invariant functions with respect to commuting transformations

  • Bálint FarkasAffiliated withFachbereich Mathematik, AG4, Technische Universität Darmstadt Email author 
  • , Szilárd Gy. RévészAffiliated withAlfréd Rényi Institute of Mathematics, Hungarian Academy of SciencesInstitut Henri Poincaré

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Summary.

As a natural generalization of various investigations in different function spaces, we study the following problem. Let A be an arbitrary non-empty set, and T j (j = 1, ..., n) be arbitrary commuting mappings from A into A. Under what conditions can we state that a function \(f : A \rightarrow {\mathbb{R}}\) is the sum of “periodic”, that is, T j -invariant functions f j ? (A function g is periodic or invariant mod T j , if \(g\, \circ\, T_j\, =\, g\).) An obvious necessary condition is that the corresponding multiple difference operator annihilates f, i.e., \(\triangle_{T_1}\ldots\triangle_{T_n}\,f\,=\,0\), where \(\triangle_{T_j}\, f :=\, f \circ T_j\,-\,f\). However, in general this condition is not sufficient, and our goal is to complement this basic condition with others, so that the set of conditions will be both necessary and sufficient.

Mathematics Subject Classification (2000).

Primary 39A10 Secondary 39B52, 39B72

Keywords.

Periodic functions periodic decomposition difference equation commuting transformations transformation invariant functions difference operator shift operator decomposition property