2005/07/29 by Bálint Farkas, Balint Farkas, Farkas, Balint +3
Mathematics · #39A10 (Primary) #39B52 #39B72 (Secondary) #Advanced Differential Equations and Dynamical Systems #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Holomorphic and Operator Theory #Mathematical Dynamics and Fractals #math.CA #msc:39A10 #msc:39B52 #msc:39B72
paper · pdf · doi:10.48550/arxiv.math/0507605
openalex publication_date 2005/07/29 · arxiv created 2007/03/12 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Tj (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, Tj-invariant functions fj? An obvious necessary condition is that the corresponding multiple difference operator annihilates f, i.e., D1 ... Dn f= 0, where Dj:=Tj-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.