A version of Hilbert's 13th problem for infinitely differentiable functions
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Hilbert's 13th problem, superposition representation of functions of several variablesAbstract
It is famous that Hilbert's 13th problem, asking if there exists a continuous real-valued function of multivariables which cannot be represented as any finite-time nested superposition of several functions of fewer variables, was proved by Kolmororov and Arnold. Actually, it is well known that there exist some other versions having been derived from the original one and still remaining to be open such as the analytic function version and the infinitely differentiable function version.
In this paper, we discuss a version of Hilbert's 13th problem for the infinitely differentiable functions. Exactly speaking, an example of an infinitely differentiable function of three real variables which cannot be represented as finite-time nested superposition of several infinitely differentiable functions of two real variables.
References
S. Akashi, A version of Hilbert’s 13th problem for analytic functions, The Bulletin of the London Mathematical Society, 35(2003), 8–14.
K. I. Babenko, Best approximations to a class of analytic functions, Izv. Akad. Nauk SSSR Ser. Mat. 22 (1958), 631–640 (Russian)
V. D. Erohin, Asymptotic theory of the ε -entropy of analytic functions, Dokl. Akad. Nauk SSSR 120 (1958), 949–952 (Russian)
E. Karapinar, On interpolative metric spaces, Filomat 38 (2024), no. 22, 7729–7734.
A. N. Kolmogorov, On the representation of continuous functions of many variables by superposition of continuous functions of one variable and addition, Dokl. Akad. Nauk SSSR 114(1957),953–956. (Russian)
V. I. Arnol’d, On functions of three variables, Dokl. Akad. Nauk SSSR 114 (1957), 679–681. (Russian)
G. G. Lorentz, Approximation of functions, Holt, Rinehart and Winston, New York, 1966.
A. G. Vitushkin, Some properties of linear superpositions of smooth functions, Dokl., 156(1964), 1003–1006.
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