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 set, and Tj (j = 1, . . . , n) be arbitrary commuting mappings – transformations – from A into A. Under what conditions can we state that a function f : A → A is the sum of “periodic”, that is, Tj-invariant functions fj? (A function g is periodic, i.e., invariant mod Tj , if g ◦ Tj = g.) An obvious necessary condition is that the corresponding multiple difference operator annihilates f , i.e., ∆T1 . . .∆Tnf = 0, where ∆Tj := Tj − I , with Tj(f) := f ◦ Tj interpreted as the usual shift operator. 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.