A superposition theorem of Kolmogorov type for bounded continuous functions

Abstract

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