A note on the representation of continuous functions by linear superpositions

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.