Year 2018, Volume 1, Issue 2, Pages 58 - 64 2018-06-01

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

Shigeo Akashi [1]

14 46

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.

Hilbert's 13th problem, superposition representation, $\varepsilon$-entropy
  • [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.
Primary Language en
Journal Section Articles
Authors

Author: Shigeo Akashi (Primary Author)
Institution: Tokyo University of Science
Country: Japan


Bibtex @research article { rna429885, journal = {Results in Nonlinear Analysis}, issn = {}, eissn = {2636-7556}, address = {Erdal KARAPINAR}, year = {2018}, volume = {1}, pages = {58 - 64}, doi = {}, title = {The existence of polynomials which are unrepresentable in Kolmogorov-Arnold superposition representation}, key = {cite}, author = {Akashi, Shigeo} }
APA Akashi, S . (2018). The existence of polynomials which are unrepresentable in Kolmogorov-Arnold superposition representation. Results in Nonlinear Analysis, 1 (2), 58-64. Retrieved from http://dergipark.org.tr/rna/issue/37067/429885
MLA Akashi, S . "The existence of polynomials which are unrepresentable in Kolmogorov-Arnold superposition representation". Results in Nonlinear Analysis 1 (2018): 58-64 <http://dergipark.org.tr/rna/issue/37067/429885>
Chicago Akashi, S . "The existence of polynomials which are unrepresentable in Kolmogorov-Arnold superposition representation". Results in Nonlinear Analysis 1 (2018): 58-64
RIS TY - JOUR T1 - The existence of polynomials which are unrepresentable in Kolmogorov-Arnold superposition representation AU - Shigeo Akashi Y1 - 2018 PY - 2018 N1 - DO - T2 - Results in Nonlinear Analysis JF - Journal JO - JOR SP - 58 EP - 64 VL - 1 IS - 2 SN - -2636-7556 M3 - UR - Y2 - 2018 ER -
EndNote %0 Results in Nonlinear Analysis The existence of polynomials which are unrepresentable in Kolmogorov-Arnold superposition representation %A Shigeo Akashi %T The existence of polynomials which are unrepresentable in Kolmogorov-Arnold superposition representation %D 2018 %J Results in Nonlinear Analysis %P -2636-7556 %V 1 %N 2 %R %U
ISNAD Akashi, Shigeo . "The existence of polynomials which are unrepresentable in Kolmogorov-Arnold superposition representation". Results in Nonlinear Analysis 1 / 2 (June 2018): 58-64.