有界连续函数的Kolmogorov型叠加定理
A superposition theorem of Kolmogorov type for bounded continuous functions
摘要：
令$C({\mathbb R}^n)$表示定义在${\mathbb R}^n$上的实值连续函数的集合。证明了对每个$n\ge 2$都存在正数$λ_1 , \ldots , λ_n$和连续函数$ϕ_1 ,\ldots , ϕ_m \in C({\mathbb R})$，且具有如下性质：对每个有界连续的$f\in C( {\mathbb R}^n )$，存在连续函数$g\in C({\mathbb R} )$，使得对每个$x=(x_1 ,\ldots , x_n )\in {\mathbb R}^n$都有$$f(x)=\sum_{q=1}^m g\left( \sum_{p=1}^n λ_p ϕ_q (x_p ) \right)$$.因此，每个$f\in C({\mathbb R}^n)$都可以从一个变量的连续函数中使用合成和加法来获得。
英文摘要：
Let $C({\mathbb R}^n)$ denote the set of real valued continuous functions defined on ${\mathbb R}^n$. We prove that for every $n\ge 2$ there are positive numbers $λ_1 , \ldots , λ_n$ and continuous functions $ϕ_1 ,\ldots , ϕ_m \in C({\mathbb R})$ with the following property: for every bounded and continuous $f\in C( {\mathbb R}^n )$ there is a continuous function $g\in C({\mathbb R} )$ such that $$f(x)=\sum_{q=1}^m g\left( \sum_{p=1}^n λ_p ϕ_q (x_p ) \right)$$ for every $x=(x_1 ,\ldots , x_n )\in {\mathbb R}^n$. Consequently, every $f\in C({\mathbb R}^n)$ can be obtained from continuous functions of one variable using compositions and additions.
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