A superposition theorem of Kolmogorov type for bounded continuous functions

2021/04/28 by Laczkovich, M. · 1 citation
#26B40 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2104.13696

Abstract

Let C(\mathbb Rⁿ) denote the set of real valued continuous functions defined on \mathbb Rⁿ. We prove that for every n≥ 2 there are positive numbers λ₁ , … , λ_n and continuous functions ϕ₁ ,… , ϕ_m ∈ C(\mathbb R) with the following property: for every bounded and continuous f∈ C( \mathbb Rⁿ ) there is a continuous function g∈ C(\mathbb R ) such that f(x)=∑_q=1^m g( ∑_p=1ⁿ λ_p ϕ_q (x_p ) ) for every x=(x₁ ,… , x_n )∈ \mathbb Rⁿ. Consequently, every f∈ C(\mathbb Rⁿ) can be obtained from continuous functions of one variable using compositions and additions.

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• Explicit Construction of Approximate Kolmogorov-Arnold Superpositions with C2-Smoothness

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