Theorem

Kolmogorov-Arnold Representation Theorem

Analysis

The Kolmogorov-Arnold representation theorem, also called the superposition theorem, states that every continuous function of several real variables on the unit cube can be written as a finite composition of continuous functions of a single variable combined only by addition. Established through the work of Andrey Kolmogorov and Vladimir Arnold, it solved a more constrained form of Hilbert's thirteenth problem as a corollary, and in this representation the inner functions are universal and independent of the particular function being represented, while the outer functions depend on it. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/

Facts

Classification

Statement Form

Existence Theorem 1

Connections

Has Statement Form

Existence Theorem

In Branch

Real Analysis

Source Kolmogorov-Arnold Representation Theorem (Wikipedia)

Proved By

Andrey Kolmogorov

Source Kolmogorov-Arnold Representation Theorem (Wikipedia)

Sources

1. Kolmogorov-Arnold Representation Theorem (Wikipedia)

Lead section
In real analysis and approximation theory, the Kolmogorov-Arnold representation theorem (or superposition theorem) states that every multivariate continuous function f : [ 0 , 1 ] n → R {\displaystyle f\colon [0,1]^{n}\to \mathbb {R} } can be represented as a superposition of continuous single-variable functions.
In Branch: Real Analysis, Lead sentence
Proved By: Andrey Kolmogorov, Lead
The works of Vladimir Arnold and Andrey Kolmogorov established that if f is a multivariate continuous function, then f can be written as a finite composition of continuous functions of a single variable and the binary operation of addition.

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Kolmogorov-Arnold Representation Theorem

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In Branch

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