On linear functional equations modulo \({\mathbb{Z}} \)
Consider a linear space \(V\), a positive integer \(n\) and real numbers \(r_0,\dots,r_{n+1}\), \(q_0,\dots,q_{n+1}\) such that \(r_iq_j\neq r_jq_i\) for \(i,j=0,\dots,n+1\), \(i\neq j\). Referring to numerous works on linear functional equations and on functional congruences, the authors consider the congruence \[ \sum_{i=0}^{n+1}\varphi_i(r_ix+q_iy)\in\mathbb{Z},\quad x,y\in V, \] where the mappings \(\varphi_i: V\to\mathbb{R}\) (\(i=0,\ldots,n+1\)) are unknown. It is proved that each of the mappings \(\varphi_i\) satisfying the above congruence satisfies also the {polynomial congruence} \[ \Delta_{y}^{n+1}\varphi_i(x)\in\mathbb{Z},\quad x,y\in V. \] Moreover, the authors study when the solutions \(\varphi_i\) are ``decent'', i.e., of the form \(\varphi_i=f_i+g_i\), where the mappings \(f_i: V\to\mathbb{R}\) satisfy the {linear functional equation} \[ \sum_{i=0}^{n+1}f_i(r_ix+q_iy)=0,\quad x,y\in V, \] and the mappings \(g_i: V\to\mathbb{Z}\) take values in \(\mathbb{Z}\).
functional equations modulo \(\mathbb{Z}\)
scientific article; zbMATH DE number 7439041
scientific article