Let $C\left({\mathbb{R}}^n\right)$ 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 ${\lambda }_1,\dots ,{\lambda }_n$ and continuous functions ${\varphi }_1,\dots ,{\varphi }_m\in C\left(\mathbb{R}\right)$ with the following property: for every bounded and continuous $f\in C\left({\mathbb{R}}^n\right)$ there is a continuous function $g\in C\left(\mathbb{R}\right)$ such that $f\left(x\right)=\sum _{q=1}^{m}g\left(\sum _{p=1}^n{\lambda }_p{\varphi }_q\left(x_p\right)\right)$ for every $x=(x_1,\dots ,x_n)\in {\mathbb{R}}^n$. Consequently, every $f\in C\left({\mathbb{R}}^n\right)$ can be obtained from continuous functions of one variable using compositions and additions.

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有界连续函数的Kolmogorov型叠加定理

让 $C\left({\mathbb{电阻}}^n\right)$ 表示定义在上的实值连续函数的集合 ${\mathbb{电阻}}^n$. 我们证明对于每个$n\ge 2$ 有正数 ${\lambda }_1,\dots ,{\lambda }_n$ 和连续函数 ${\phi }_1,\dots ,{\phi }_{米}\in C\left(\mathbb{电阻}\right)$ 具有以下性质：对于每一个有界和连续的 $F\in C\left({\mathbb{电阻}}^n\right)$ 有一个连续函数 $G\in C\left(\mathbb{电阻}\right)$ 以至于 $F\left(X\right)=\sum _{q=1}^{米}G\left(\sum _{磷=1}^n{\lambda }_{磷}{\phi }_q\left(X_{磷}\right)\right)$ 对于每个 $X=(X_1,\dots ,X_n)\in {\mathbb{电阻}}^n$. 因此，每$F\in C\left({\mathbb{电阻}}^n\right)$ 可以使用组合和加法从一个变量的连续函数中获得。