The existence of polynomials which are unrepresentable in Kolmogorov-Arnold superposition representation

Yıl 2018, Cilt: 1 Sayı: 2, 58 - 64, 31.08.2018

Shigeo Akashi

Öz

In this paper, it is proved that there exist polynomials of three complex variables which cannot be represented as any Kolmogorov-Arnold superposition, which has played important roles in the original version of Hilbert's 13th problem.

Anahtar Kelimeler

Hilbert's 13th problem , superposition representation , $\varepsilon$-entropy

Kaynakça

• [1] S. Akashi, A version of Hilbert’s 13th problem for analytic functions, The Bulletin of the London Mathematical Society, 35(2003), 8-14.
• [2] K. I. Babenko, On the best approximation of a class of analytic functions, Izv. 22(1958), 631-640.
• [3] V. D. Erohin, On the asymptotic behavior of the ε-entropy of analytic functions, Dokl., 120(1958), 949-952.
• [4] G. G. Lorentz, Approximation of functions, Holt, Rinehart and Winston, New York, 1966.
• [5] A. N. Kolmogorov, On the representation of continuous functions of several variables by superpositions of continuous functions of one variable and addition, Dokl., 114(1957), 679-681.
• [6] S.N. Mergelyan, On the representation of functions by series of polynomials on closed sets, Transl. Amer. Math. Soc., 3 (1962), 287-293.
• [7] S.N. Mergelyan, Uniform approximation to functions of a complex variable, Transl. Amer. Math. Soc., 3 (1962), 294391. [8] A. G. Vitushkin, Some properties of linear superpositions of smooth functions, Dokl., 156(1964), 1003-1006.