Linear functional equations on the line (Q1891855)

The functional equation \(\sum^ L_{j= 1} A_ j(x) y(F_ j x)= \gamma(x)\), where \(A_ j: \mathbb{R}\to \text{Hom}(\mathbb{C}^ m, \mathbb{C}^ n)\) and \(\gamma: \mathbb{R}\to \mathbb{C}^ n\) are given \(C^ k\)-maps and \(F_ j: \mathbb{R}\to \mathbb{R}\) are given \(C^ k\)-diffeomorphisms, is considered. A general theory of this equation via dynamical properties of the system \(\{F_ j\}\) is built. Results on the reduction of the global \(C^ k\)- solvability problem to the local one are given.