Note on decomposition of bounded functions into the sum of periodic terms (Q1207390)

Let \(T\) be a transformation on a set \(A\). A real-valued function \(g\) on \(A\) is called \(T\)-periodic, if \(\Delta_T g:= g\circ T-g=0\). A class \({\mathcal F}\) of real-valued functions on \(A\) has the decomposition property with respect to \(T_1,\dots, T_k\) if for every \(f\in{\mathcal F}\) with \(\Delta_{T_1}\dots \Delta_{T_k} f=0\) there exist \(T_i\)-periodic functions \(f_i\in{\mathcal F}\) such that \(f= f_1+\dots+ f_k\). It is shown, that if the family of all the iterates \(T_i^n\) of commuting transformations is equicontinuous, then the class of uniformly continuous bounded functions has the decomposition property with respect to \(T_1,\dots, T_k\).