A superposition theorem of Kolmogorov type for bounded continuous functions

TL;DR

This paper establishes a superposition theorem for bounded continuous functions on Euclidean spaces, showing they can be represented as sums of compositions of univariate continuous functions, extending Kolmogorov's superposition concept.

Contribution

It proves a new superposition theorem for bounded continuous functions, providing explicit conditions and constructions for representing multivariate functions via univariate functions.

Findings

Every bounded continuous function on R^n can be expressed as a finite sum of compositions of univariate continuous functions.

The theorem generalizes Kolmogorov's superposition theorem to bounded functions on Euclidean spaces.

Explicit parameters and functions are constructed to achieve the representation.

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.