Abstract
In 1957, Kolmogorov and Arnold gave a solution to the 13th problem which had been formulated by Hilbert in 1900. Actually, it is known that there exist many open problems which can be derived from the original problem. From the function-theoretic point of view, Hilbert's 13th problem can be exactly characterized as the superposition representability problem for continuous functions of several variables. In this paper, the solution to the superposition representability problem for infinitely differentiable functions of several variables is given.
Highlights
In 1964, Vitushkin 2 solved the problem, which had been derived from Hilbert’s 13th problem, asking if all finitely differentiable realvalued functions of several real variables can be represented as superpositions of finitely differentiable functions of fewer variables
Let F P 3 resp., F P 2 be a set of functions of three variables resp., two variables satisfying the condition P such as continuity or differentiability
Strong representabity: there exists a positive integer k satisfying that, for any function f of F P 3, f can be represented as a k-time nested superposition constructed from 2k 1 − 1 functions of F P 2
Summary
In 1957, Kolmogorov and Arnold 1 solved Hilbert’s 13th problem asking if all continuous real-valued functions of several real variables can be represented as superpositions of continuous functions of fewer variables. Strong representabity: there exists a positive integer k satisfying that, for any function f of F P 3, f can be represented as a k-time nested superposition constructed from 2k 1 − 1 functions of F P 2. Weak representability: for any function f of F P 3, there exists a positive integer kf such that f can be represented as a kf -time nested superposition constructed from several functions of F P 2. Strong irrepresentability: there exists a function f of F P 3 which cannot be represented as any finite-time nested superposition constructed from several functions of. Weak irrepresentability: for any positive integer k, there exists a function fk of F P 3 which cannot be represented as any k-time nested superposition constructed from several functions of F P 2. As for the nonlinear theoretic approximation methods, we can refer to Takahashi’s results 8
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