On rational Abel – Poisson means on a segment and approximations of Markov functions

  • Pavel G. Patseika Yanka Kupala State University of Grodno, 22 Ažeška Street, Hrodna 230023, Belarus https://orcid.org/0000-0001-7835-0500
  • Yauheni A. Rouba Yanka Kupala State University of Grodno, 22 Ažeška Street, Hrodna 230023, Belarus
DOI: https://doi.org/10.33581/2520-6508-2021-3-6-24

Abstract

Approximations on the segment [−1, 1] of Markov functions by Abel – Poisson sums of a rational integral operator of Fourier type associated with the Chebyshev – Markov system of algebraic fractions in the case of a fixed number of geometrically different poles are investigated. An integral representation of approximations and an estimate of uniform approximations are found. Approximations of Markov functions in the case when the measure µ satisfies the conditions suppµ = [1, a], a > 1, dµ(t) = φ(t)dt and φ(t) ≍ (t − 1)α on [1, a], a are studied and estimates of pointwise and uniform approximations and the asymptotic expression of the majorant of uniform approximations are obtained. The optimal values of the parameters at which the majorant has the highest rate of decrease are found. As a corollary, asymptotic estimates of approximations on the segment [−1, 1] are given by the method of rational approximation of some elementary Markov functions under study.

Author Biographies

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

PhD (physics and mathematics); associate professor 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

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Published
2021-11-19
Keywords: Markov functions, rational integral operators, Abel – Poisson means, Chebyshev – Markov algebraic fractions, best approximations, asymptotic estimates, exact constants
How to Cite
Patseika, P. G., & Rouba, Y. A. (2021). On rational Abel – Poisson means on a segment and approximations of Markov functions. Journal of the Belarusian State University. Mathematics and Informatics, 3, 6-24. https://doi.org/10.33581/2520-6508-2021-3-6-24
Issue
No 3 (2021): Journal of the Belarusian State University. Mathematics and Informatics
Section
Real, Complex and Functional Analysis

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