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.
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Source Kolmogorov-Arnold Representation Theorem (Wikipedia)
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Source Kolmogorov-Arnold Representation Theorem (Wikipedia)
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1. Kolmogorov-Arnold Representation Theorem (Wikipedia)
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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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