  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.

Classical Analysis and ODEs