A structured proof of the Kolmogorov superposition theorem

05/02/2021

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by   S. Dzhenzher, et al.

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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 φ_1,…,φ_5 : [ 0, 1 ]→ [ 0,1 ] such that for any continuous function f: [ 0,1 ]^2→ℝ there is a continuous function h: [ 0,3 ]→ℝ such that for any x,y∈ [ 0, 1 ] we have f(x,y)=∑_k=1^5 h(φ_k(x)+√(2) φ_k(y)). The proof is accessible to non-specialists, in particular, to students familiar with only basic properties of continuous functions.