On the representation by linear superpositions

TL;DR

This paper establishes a practical necessary and sufficient condition for representing arbitrary functions as linear superpositions, extending previous results and leading to an analogue of Kolmogorov's superposition theorem.

Contribution

It provides a new, topology-free criterion for function representation by linear superpositions, generalizing prior work and connecting to Kolmogorov's theorem.

Findings

Necessary and sufficient condition for representability of functions by linear superpositions.

Representation for continuous functions implies representation for all functions.

Extension of Kolmogorov superposition theorem to multivariate functions.

Abstract

In a number of papers, Y. Sternfeld investigated the problems of representation of continuous and bounded functions by linear superpositions. In particular, he proved that if such representation holds for continuous functions, then it holds for bounded functions. We consider the same problem without involving any topology and establish a rather practical necessary and sufficient condition for representability of an arbitrary function by linear superpositions. In particular, we show that if some representation by linear superpositions holds for continuous functions, then it holds for all functions. This will lead us to the analogue of the well-known Kolmogorov superposition theorem for multivariate functions on the $d$-dimensional unit cube.