A structured proof of Kolmogorov's Superposition Theorem
This work clarifies a foundational mathematical theorem for researchers and students, but it is incremental as it repackages an existing proof.
The paper provides a structured exposition of a known proof for Kolmogorov's Superposition Theorem, which solves Hilbert's 13th problem by representing any continuous function of two variables as a sum of compositions of continuous functions, making the proof accessible to non-specialists like students.
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 $\varphi_1,\ldots,\varphi_5 : [\,0, 1\,]\to [\,0,1\,]$ 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 $$f(x,y)=\sum\limits_{k=1}^5 h\left(\varphi_k(x)+\sqrt{2}\,\varphi_k(y)\right).$$ The proof is accessible to non-specialists, in particular, to students familiar with only basic properties of continuous functions.