On functional equations associated with characters of unitary representations of groups

Abstract

This is mainly concerned with the following functional equation: $$f\left( x \right)f\left( y \right) = \frac{1}{{\left| \Phi \right|}}\sum\limits_{\varphi \in \Phi } {M_t \left( {f\left( {xt\varphi \left( y \right)t^{ - 1} } \right)} \right)} , x,y \in G,$$ (*) whereG is a topological group,M stands for the invariant mean on the space of almost periodic functions defined onG and Ф is a finite set of morphisms ofG. It is shown that almost periodic solutions of this equation can be expressed by means of characters of finite-dimensional irreducible unitary representations of the groupG. In the case whereG is compact, we study an integral version of Eq. (*) and we determine its solutions in the space\(\mathfrak{L}_1 \)(G). Our considerations are motivated by the results of H. Weyl and A. Weil on integrable solutions of the equation $$f\left( x \right)f\left( y \right) = \int_G f \left( {xtyt^{ - 1} } \right)dt, x,y \in G$$ .