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The Kac–Bernstein functional equation on Abelian groups

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Abstract

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.

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Acknowledgements

I would like to thank the first referee for the attention to my article and language corrections. My special thanks to the second referee for carefully reading of my article and many useful suggestions, in particular, for Remark 2.1 which I included in the article.

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  1. Kharkiv, Ukraine

    Gennadiy Feldman

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  1. Gennadiy Feldman
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Correspondence to Gennadiy Feldman.

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Feldman, G. The Kac–Bernstein functional equation on Abelian groups. Aequat. Math. 95, 737–750 (2021). https://doi.org/10.1007/s00010-021-00787-w

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