A note on the representation of continuous functions by linear superpositions

TL;DR

This paper investigates the conditions under which continuous functions can be represented as linear superpositions of continuous functions composed with fixed functions, providing a necessary condition for such representations.

Contribution

It introduces a necessary condition for representing continuous functions as linear superpositions, extending previous work that mainly focused on sufficient conditions for the case k=2.

Findings

Provides a necessary condition for the representation of continuous functions by linear superpositions.

Extends understanding of the representation problem for cases where k>2.

Builds on prior sufficient conditions by offering a complementary necessary condition.

Abstract

We consider the problem of the representation of real continuous functions by linear superpositions $\sum_{i=1}^{k}g_{i}\circ p_{i}$ with continuous $g_{i}$ and $p_{i}$. This problem was considered by many authors. But complete, and at the same time explicit and practical solutions to the problem was given only for the case $k=2$. For $k>2$, a rather practical sufficient condition for the representation can be found in Sternfeld [17] and Sproston, Strauss [16]. In this short note, we give a necessary condition of such kind for the representability of continuous functions.