Solution of several functional equations on abelian groups with involution

Abstract

Let G be a locally compact abelian Hausdorff group, let $\sigma$ be a continuous involution on G, and let $\mu ,\nu$ be regular, compactly supported, complex-valued Borel measures on G. We determine the continuous solutions $f,g:G\rightarrow {\mathbb {C}}$ of each of the two functional equations

$$\begin{aligned}&\int _{G}f(x+y+t)d\mu (t)+\int _{G}f(x+\sigma (y)+t)d\nu (t)=f(x)g(y),\quad x,y\in G,\\&\int _{G}f(x+y+t)d\mu (t)+\int _{G}f(x+\sigma (y)+t)d\nu (t)=g(x)f(y),\quad x,y\in G, \end{aligned}$$

in terms of characters and additive functions. These equations provides a common generalization of many functional equations such as d’Alembert’s, Cauchy’s, Gajda’s, Kannappan’s, Van Vleck’s, or Wilson’s equations. So, several functional equations will be solved.