Decomposition as the sum of invariant functions with respect to   commuting transformations
  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. 
Balint Farkas and Szilard Revesz