A structured proof of Kolmogorov's Superposition Theorem

Abstract

We present a well-structured detailed exposition of a well-known proof of the following celebrated result solving Hilbert's 13th problem on superpositions. For functions of 2 variables the statement is as follows. Kolmogorov Theorem. There are continuous functions ${\phi }_1,\dots ,{\phi }_5:[0,1]\to [0,1]$ such that for any continuous function $f:[0,1{]}^2\to \mathbb{R}$ there is a continuous function $h:[0,3]\to \mathbb{R}$ such that for any $x,y\in [0,1]$ we have $$f(x,y)=\sum _{k=1}^{5}h\left({\phi }_k(x)+\sqrt{2}{\phi }_k(y)\right).$$ The proof is accessible to non-specialists, in particular, to students familiar with only basic properties of continuous functions.

English: 7+8 pages, 1+1 figures, in English and in Russian; exposition improved, Russian version added