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 $\lambda _1 , \ldots , \lambda _n$ and continuous functions $\phi_1 ,\ldots , \phi _m \in C({\mathbb R})$ 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 $$f(x)=\sum_{q=1}^m g\left( \sum_{p=1}^n \lambda _p \phi _q (x_p ) \right)$$ 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.