Convolution-type functional equations

The pioneer work of János Aczél on functional equations led to a comprehensive theory on this field. His basic book “Lectures on functional equations and their applications” served as a cornerstone of this theory. A great number of classical functional equations are of or can be reduced to convolution-type systems of functional equations. It turned out that this class can effectively be studied using the methods of harmonic analysis and spectral synthesis. In this paper we give a short summary starting from the basics of the theory to the present state and main results.