On the approximation of conjugate functions and their derivatives on the segment by partial sums of Fourier – Chebyshev series

  • Pavel G. Patseika Yanka Kupala State University of Grodno, 22 Azheshka Street, Grodna 230023, Belarus https://orcid.org/0000-0001-7835-0500
  • Yauheni A. Rouba Yanka Kupala State University of Grodno, 22 Azheshka Street, Grodna 230023, Belarus
  • Kanstantin A. Smatrytski Yanka Kupala State University of Grodno, 22 Azheshka Street, Grodna 230023, Belarus

Abstract

In this paper, we study the approximation of conjugate functions with the density f є H (α)[–1, 1], α є (0, 1], on the segment [–1, 1], by the conjugate Fourier – Chebyshev series. We establish the order estimations of the approximation depending on the location of a point on the segment. It is noted that approximation at the endpoints of the segment has a higher rate of decrease in comparison with the whole segment. We introduce classes of functions, which, in a certain sense, can be associated with the derivative of a conjugate function on the segment [–1, 1], and the approximation of functions from these classes by partial sums of the Fourier – Chebyshev series is studied. An integral representation of the approximation is found. In the case when the density f є W1H (α)[–1, 1], α є (0, 1], the order estimations of the approximation, depending on the location of the point on the segment, are established. The case, when the density f (t) = |t|s, s > 1 is considered. In this case, an integral representation of the approximation, estimations for pointwise and uniform approximations, as well as an asymptotic estimation for the uniform approximation are obtained. It is noted that the order of the uniform approximations of the function under study by partial sums of the Fourier – Chebyshev series and the corresponding conjugate function by conjugate sums coincide.

Author Biographies

Pavel G. Patseika, Yanka Kupala State University of Grodno, 22 Azheshka Street, Grodna 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 Azheshka Street, Grodna 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 Azheshka Street, Grodna 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
2024-07-24
Keywords: singular integral on a segment, conjugate function, Lipschitz condition, Fourier – Chebyshev series, uniform estimations, asymptotic estimations
Supporting Agencies The authors would like to express their sincere gratitude to full professor, doctor of science (physics and mathematics) A. Pekarskii for a number of valuable comments and advice, which were taken into account in the final edition of the paper.
How to Cite
Patseika, P. G., Rouba, Y. A., & Smatrytski, K. A. (2024). On the approximation of conjugate functions and their derivatives on the segment by partial sums of Fourier – Chebyshev series. Journal of the Belarusian State University. Mathematics and Informatics, 2, 6-18. Retrieved from https://journals.bsu.by/index.php/mathematics/article/view/5231
Issue
No 2 (2024): Journal of the Belarusian State University. Mathematics and Informatics
Section
Real, Complex and Functional Analysis

Copyright (c) 2024 Journal of the Belarusian State University. Mathematics and Informatics

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