Aequationes mathematicae

, Volume 73, Issue 3, pp 233–248 | Cite as

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



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 


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


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Copyright information

© Birkhäuser Verlag, Basel 2007

Authors and Affiliations

  1. 1.Fachbereich Mathematik, AG4Technische Universität DarmstadtDarmstadtGermany
  2. 2.Alfréd Rényi Institute of MathematicsHungarian Academy of SciencesBudapestHungary
  3. 3.Institut Henri PoincaréParisFrance

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