On one rational integral operator of Fourier – Chebyshev type and approximation of Markov functions

DOI: https://doi.org/10.33581/2520-6508-2020-2-6-27

Abstract

The purpose of this paper is to construct an integral rational Fourier operator based on the system of Chebyshev – Markov rational functions and to study its approximation properties on classes of Markov functions. In the introduction the main results of well-known works on approximations of Markov functions are present. Rational approximation of such functions is a well-known classical problem. It was studied by A. A. Gonchar, T. Ganelius, J.-E. Andersson, A. A. Pekarskii, G. Stahl and other authors. In the main part an integral operator of the Fourier – Chebyshev type with respect to the rational Chebyshev – Markov functions, which is a rational function of order no higher than n is introduced, and approximation of Markov functions is studied. If the measure satisfies the following conditions: suppμ = [1, a], a > 1, dμ(t) = ϕ(t)dt and ϕ(t) ἆ (t − 1)α on [1, a] the estimates of pointwise and uniform approximation and the asymptotic expression of the majorant of uniform approximation are established. In the case of a fixed number of geometrically distinct poles in the extended complex plane, values of optimal parameters that provide the highest rate of decreasing of this majorant are found, as well as asymptotically accurate estimates of the best uniform approximation by this method in the case of an even number of geometrically distinct poles of the approximating function. In the final part we present asymptotic estimates of approximation of some elementary functions, which can be presented by Markov functions.

Author Biographies

Pavel G. Patseika, Yanka Kupala State University of Grodno, 22 Ažeška Street, Hrodna 230023, Belarus

postgraduate student at the department of fundamental and applied mathematics, faculty of mathematics and informatics

Yauheni A. Rouba, Yanka Kupala State University of Grodno, 22 Ažeška Street, Hrodna 230023, Belarus

doctor of science (physics and mathematics), full professor; head of the department of fundamental and  applied mathematics, faculty of mathematics and informatics

Kanstantin A. Smatrytski, Yanka Kupala State University of Grodno, 22 Ažeška Street, Hrodna 230023, Belarus

PhD (physics and mathematics), docent; associate professor at the department of fundamental and applied mathematics, faculty of mathematics and informatics

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Published
2020-07-30
Keywords: Markov function, integral rational operator of Fourier type, Chebyshev – Markov rational function, majorant of uniform approximation, asymptotic estimate, best approximation, exact constant
How to Cite
Patseika, P. G., Rouba, Y. A., & Smatrytski, K. A. (2020). On one rational integral operator of Fourier – Chebyshev type and approximation of Markov functions. Journal of the Belarusian State University. Mathematics and Informatics, 2, 6-27. https://doi.org/10.33581/2520-6508-2020-2-6-27
Issue
No 2 (2020): Journal of the Belarusian State University. Mathematics and Informatics
Section
Real, Complex and Functional Analysis

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