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

Abstract

Let A be an arbitrary set. For any transformation T (self-map of A) let T(f)(x):=f(T(x)) (for all x in A) be the usual shift operator. A function g is called periodic, i.e., invariant mod T, if Tg=g (=Ig, where I is the identity operator). As a natural generalization of various earlier investigations in different function spaces, we study the following problem. Let T_j (j=1,...,n) be arbitrary commuting mappings -- transformations -- from A into A. Under what conditions can we state that a function f from A to A is the sum of "periodic", that is, T_j-invariant functions f_j? An obvious necessary condition is that the corresponding multiple difference operator annihilates f, i.e., D_1 ... D_n f= 0, where D_j:=T_j-I. 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.