The solution of the functional equation of D'Alembert's type for commutative groups

Abstract

A functional equation of the form ϕ1(x+y)+ϕ2(x−y)=∑inαi(x)βi(y), where functions ϕ1,ϕ2,αi,βi, i=1,…,n are defined on a commutative group, is solved. We also obtain conditions for the solutions of this equation to be matrix elements of a finite dimensional representation of the group.