On the representation of functions of several variables as a superposition of functions of a smaller number of variables

Abstract

In this paper we wish to give an account of several recent papers by Moscow mathematicians devoted to the question in the title of this paper. §1 contains the definition of superposition of functions and the statement of Hilbert’s 13th problem relating to superpositions. §2 is devoted to superpositions of smooth functions. In §3 we present several very recent papers, in spite of the fact that the content of that section is now perhaps only of historical interest. The principal topic there is the description given by Kronrod of “the tree of components of a function of several variables”, which is a concept whose popularization would seem to be very desirable (although the connection between this concept and the problems considered in our paper has proved to be less close than it originally appeared). The reader interested only in the strongest (and, moreover, the simplest in its method of proof) result relating to the representation of continuous functions of several variables as superpositions of functions of a smaller number of variables can, after looking at the introductory §1 go straight to §4, missing out §2–3. In addition, the smaller print in this paper means, as usual, that the corresponding material is auxiliary and omitting it will not affect the reader’s understanding of what follows.