We consider the Kac–Bernstein functional equation

$$\begin{aligned} f(x+y)g(x-y)=f(x)f(y)g(x)g(-y), \quad x, y\in X, \end{aligned}$$

on an arbitrary Abelian group X. We give a complete description of the solutions of this equation in the class of positive functions. This result is a generalization of a theorem by Pl. Kannappan. We also study the solutions of this equation in the class of complex-valued Hermitian functions.

中文翻译：

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关于Abelian群的Kac–Bernstein函数方程

我们考虑了Kac–Bernstein函数方程

$$ \ begin {aligned} f（x + y）g（xy）= f（x）f（y）g（x）g（-y），\ quad x，y \ in X，\ end {aligned} $$

在任意阿贝尔群X上。我们在正函数类中给出对该方程解的完整描述。该结果是P1对一个定理的推广。Kannappan。我们还在复值Hermitian函数类中研究了该方程的解。