, Volume 73, Issue 3, pp 233-248

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

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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.

Manuscript received: July 18, 2005 and, in final form, March 13, 2006.