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Non-existence of measurable solutions of certain functional equations via probabilistic approaches
Aequationes Mathematicae ( IF 0.8 ) Pub Date : 2021-05-26 , DOI: 10.1007/s00010-021-00811-z Kazuki Okamura
中文翻译:
通过概率方法不存在某些函数方程的可测解
更新日期:2021-05-26
Aequationes Mathematicae ( IF 0.8 ) Pub Date : 2021-05-26 , DOI: 10.1007/s00010-021-00811-z Kazuki Okamura
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.
中文翻译:
通过概率方法不存在某些函数方程的可测解
本文以\(f(x)+ g(y)= h(x,y)\)的形式处理函数方程,其中h是给定的,而f和g是未知的。我们将证明,如果h是与均匀分布或柯西分布的特征相关的Borel可测量函数,则该方程式没有可测量的解。我们的证明使用狄拉克测度的特征,它也适用于arctan方程。