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

TL;DR

This paper characterizes all positive solutions to the Kac--Bernstein functional equation on Abelian groups and explores solutions among non-vanishing complex functions satisfying Hermitian conditions.

Contribution

It provides a complete description of solutions in the class of positive functions and extends analysis to non-vanishing complex functions with Hermitian properties.

Findings

Explicit form of positive solutions on Abelian groups

Solutions in the class of non-vanishing Hermitian functions

Extension of solutions to complex-valued 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.