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arXiv · 2105.09804

Non-existence of measurable solutions of certain functional equations via probabilistic approaches

Kazuki Okamura
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Abstract

This paper deals with functional equations in the form of $f(x) + g(y) = h(x,y)$ where $h$ is given and $f$ and $g$ are unknown. We will show that if $h$ is a Borel measurable function associated with characterizations of the uniform or Cauchy distributions, then there is no measurable solutions of the equation. Our proof uses a characterization of the Dirac measure and it is also applicable to the arctan equation.

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Kazuki Okamura. 2021-05-20. Non-existence of measurable solutions of certain functional equations via probabilistic approaches. https://doi.org/10.1007/s00010-021-00811-z

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