Rational interpolation of a function |x|^α with Chebyshev – Markov nodes of the first kind

  • Yauheni A. Rouba Yanka Kupala State University of Grodno, 22 Ažeška Street, Hrodna 230023, Belarus
  • Victoria Yu. Medvedeva Yanka Kupala State University of Grodno, 22 Ažeška Street, Hrodna 230023, Belarus
DOI: https://doi.org/10.33581/2520-6508-2023-1-6-19

Abstract

This paper considers the approximations of the function |x|^α, α > 0, by interpolation rational Lagrange functions on the interval [−1, 1]. Zeros of the rational Chebyshev – Markov function of the first kind are chosen as interpolation nodes. An integral representation of the interpolation remainder and an upper estimation for the considered uniform approximations are obtained. The polynomial and general rational cases are studied in detail. In the polynomial case, an asymptotic estimate for uniform approximations is found. When approximating by interpolation rational Lagrange functions with Chebyshev – Markov nodes of the first kind, the upper and lower estimations are found. These estimations are close to that of the best uniform approximations of the function under consideration on the interval [−1, 1].

Author Biographies

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

Victoria Yu. Medvedeva, 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

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Published
2023-03-16
Keywords: rational Chebyshev – Markov fraction, rational interpolation, function with power singularity
Supporting Agencies The authors are grateful to professor A. A. Pekarsky for a useful discussion of the results of this work.
How to Cite
Rouba, Y. A., & Medvedeva, V. Y. (2023). Rational interpolation of a function |x|^α with Chebyshev – Markov nodes of the first kind. Journal of the Belarusian State University. Mathematics and Informatics, 1, 6-19. https://doi.org/10.33581/2520-6508-2023-1-6-19
Issue
No 1 (2023): Journal of the Belarusian State University. Mathematics and Informatics
Section
Real, Complex and Functional Analysis

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