Solution of the Kac--Bernstein functional equation on Abelian groups in the class of positive functions

Abstract

The general form of the solutions of the Kac--Bernstein functional equation $$f(x+y)g(x-y)=f(x)f(y)g(x)g(-y),x,y\in X,$$ on an arbitrary Abelian group $X$ in the class of positive functions is obtained. We also study the solutions of this equation in the class of complex-valued functions that do not vanish and satisfy the Hermitian condition.