Fejer means of rational Fourier – Chebyshev series and approximation of function |x|s

DOI: https://doi.org/10.33581/2520-6508-2019-3-18-34

Abstract

Approximation properties of Fejer means of Fourier series by Chebyshev – Markov system of algebraic fractions and approximation by Fejer means of function |x|s, 0 < s < 2, on the interval [−1,1], are studied. One orthogonal system of Chebyshev – Markov algebraic fractions is considers, and Fejer means of the corresponding rational Fourier – Chebyshev series is introduce. The order of approximations of the sequence of Fejer means of continuous functions on a segment in terms of the continuity module and sufficient conditions on the parameter providing uniform convergence are established. A estimates of the pointwise and uniform approximation of the function |x|s, 0 < s < 2, on the interval [−1,1], the asymptotic expressions under n→∞ of majorant of uniform approximations, and the optimal value of the parameter, which provides the highest rate of approximation of the studied functions are sums of rational use of Fourier – Chebyshev are found. 

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

References

  1. Fejér L. Untersuchungen über Fouriersche Reihen. Mathematische Annalen. 1904;58(1–2):51–69. DOI: 10.1007/BF01447779.
  2. Lebesgue H. Sur les intégrales singulières. Annales de la faculté des sciences de Toulouse. 3e série. 1909;1:25–117.
  3. Bernstein S. Sur l’ordre de la meilleure approximation des fonctions continues par des polynomes de degré donné. Bruxelles: Hayez, imprimeur des Academies Royales; 1912. 104 p.
  4. Nikol’skii SM. [On the asymptotic behavior of the remainder under approximation of functions satisfying the Lipschitz condition, by fejér sums]. Izvestiya AN SSSR. Seriya matematicheskaya. 1940;4(6):501–508. Russian.
  5. Zygmund A. On the degree of approximation of functions by Fejer means. Bulletin of the American Mathematical Society. 1945;51(4):274 –278.
  6. Novikov OA, Rovenska OG. Approximation of classes of Poisson integrals by Fejer sums. Komp’yuternye issledovaniya i modelirovanie. 2015;7(4):813–819. Russian. DOI: 10.20537/2076-7633-2015-7-4-813-819.
  7. Efimov AV. [On the approximation of some classes of continuous functions by Fourier sums and Fejer sums]. Izvestiya AN SSSR. Seriya matematicheskaya. 1958;22(1):81–116. Russian.
  8. Lebed’ GK, Avdeenko AA. [On the approximation of periodic functions by Fejér sums]. Izvestiya AN SSSR. Seriya matematicheskaya. 1971;35(1):83–92. Russian.
  9. Dzhrbashyan MM. [On the theory of Fourier series on rational functions]. Izvestiya Akademii nauk Armyanskoi SSR. Seriya fiziko-matematicheskaya. 1956;9(7):1–27. Russian.
  10. Rusak VN. Ratsional’nye funktsii kak apparat priblizheniya [Rational functions as an apparatus of approximation]. Minsk: Belorusskii gosudarstvennyi universitet im. V. I. Lenina; 1979. 179 p. Russian.
  11. Petrushev PP, Popov VA. Rational approximation of real functions. Cambridge: Cambridge University Press; 1987. 386 p.
  12. Rusak VN. [Sharp order estimates for best rational approximations in classes of functions representable as convolutions]. Doklady Akademii nauk SSSR. 1984;279(4):810 – 812. Russian.
  13. Rusak VN. [Exact order estimates for best rational approximations in classes of functions representable as convolution]. Matematicheskii sbornik. 1985;128(4):492–515. Russian.
  14. Pekarskii AA. [Chebyshev rational approximations in a circle, on a circle, and on a segment]. Matematicheskii sbornik. 1987; 133(1):86 –102. Russian.
  15. Smotritskii KA. [Approximation by rational operators of Valle Poussin on a segment]. Trudy Instituta matematiki NAN Belarusi. 2001;9:136 –139. Russian.
  16. Rovba EA. Rational integral operators on a segment. Vestnik BGU. Seriya 1. Fizika. Matematika. Informatika. 1996;1:34 –39. Russian.
  17. Smotrytski KA. On the approximation of the convex functions by rational integral operators on the segment. Vestnik BGU. Seriya 1. Fizika. Matematika. Informatika. 2005;3:64 –70. Russian.
  18. Rovba EA. [Approximation of functions differentiable in the Riemann – Liouville sense by rational operators]. Doklady of the National Academy of Sciences of Belarus. 1996;40(6):18–22. Russian.
  19. Rovba EA. [On the approximation by rational Fejer and Jackson operators of bounded variation functions]. Doklady of the National Academy of Sciences of Belarus. 1998;42(4):13–17. Russian.
  20. Bernstein S. Sur la meilleure approximation de x par des polynomes de degres donnes. Acta Mathematica. 1914;37:1–57. DOI: 10.1007/BF02401828.
  21. Newman DJ. Rational approximation to x. Michigan Mathematical Journal. 1964;11(1):11–14. DOI: 10.1307/mmj/1028999029.
  22. Bulanov AP. [Asymptotics for least deviation of x by rational functions]. Matematicheskii sbornik. 1968;76(118-2):288–303. Russian.
  23. Vyacheslavov NS. [On the approximation of the function x by rational functions]. Matematicheskie zametki. 1974;16(1): 163–171. Russian.
  24. Shtal’ G. [Best uniform rational approximation of x on −[] 11 ,. Matematicheskii sbornik. 1992;183(8):85–118. Russian.
  25. Bernstein SN. [Sur la meilleure approximation de x p par des polynomes de degres tres eleves]. Izvestiya AN SSSR. Seriya matematicheskaya. 1938;2(2):169 –190. Russian.
  26. Freud G, Szabados J. Rational approximation to x a. Acta Mathematica Academiae Scientiarum Hungaricae. 1967;18(3– 4): 393–399. DOI: 10.1007/BF02280298.
  27. Gonchar AA. [On the rate of rational approximation of continuous functions with characteristic features]. Matematicheskii sbor nik. 1967;73(4):630 – 638. Russian.
  28. Vyacheslavov N. [On the approximation of xa by rational functions]. Izvestiya AN SSSR. Seriya matematicheskaya. 1980;44(1): 92–109. Russian.
  29. Stahl HR. Best uniform rational approximation of x a on [0, 1]. Bulletin of the American Mathematical Society. 1993;28(1):116 –122.
  30. Revers M. On the asymptotics of polynomial interpolation to x a at the Chebyshev nodes. Journal of Approximation Theory. 2013;65:70 – 82. DOI: 10.1016/j.jat.2012.09.005.
  31. Raitsin RA. [Asymptotic properties of uniform approximations of functions with algebraic features by partial sums of the Fourier – Chebyshev series]. Izvestiya vysshikh uchebnykh zavedenii. Matematika. 1980;3:45– 49. Russian.
  32. Rouba Y, Patseika P, Smatrytski K. On one system of rational Chebyshev – Markov fractions. Analysis Mathematica. 2018; 44(1):115–140. DOI: 10.1007/s10476-018-0110-7.
  33. Natanson IP. Konstruktivnaya teoriya funktsii [Constructive theory of functions]. Moscow: GITTL; 1949. 684 p. Russian.
  34. Timan AF. Teoriya priblizhenii funktsii deistvitel’nogo peremennogo [Theory of approximations of functions of a real variable]. Moscow: GIFML; 1960. 624 p. Russian.
  35. Titchmarsh E. Teoriya funktsii [Theory of functions]. Moscow: Nauka; 1980. 463 p. Russian.
  36. Rovba EA, Potseiko PG. [Approximation of the function x s on the segment [−1,1] by partial sums of the rational Fourier – Chebyshev series]. Vesnik Grodzenskaga dzjarzhawnaga wniversitjeta imja Janki Kupaly. Seryja 2. Matjematyka. Fizika. Infarmatyka, vylichal’naja tjehnika i kiravanne. 2019;9(3):16 –28. Russian.
  37. Sidorov YuV, Fedoryuk MV, Shabunin MI. Lektsii po teorii funktsii kompleksnogo peremennogo [Lectures on the theory of functions of a complex variable]. Moscow: Nauka; 1989. 480 p. Russian.
  38. Evgrafov MA. Asimptoticheskie otsenki i tselye funktsii [Asymptotic estimates and entire functions]. Moscow: Nauka; 1979. 320 p. Russian.
  39. Fedoryuk MV. Asimptotika. Integraly i ryady [Asymptotics. Integrals and series]. Moscow: Nauka; 1987. 544 p. Russian.
  40. Copson ET. Asymptotic Expansions. Cambridge: Cambridge University Press; 1965. 124 p. (Cambridge Tracts in Mathematics and Mathematical Physics; no. 55).
Published
2019-11-27
Keywords: Fourier – Chebyshev series, partial sums, Fejer means, modulus of continuity, uniform convergence, asymptotic estimates, exact constants
How to Cite
Patseika, P. G., & Rouba , Y. A. (2019). Fejer means of rational Fourier – Chebyshev series and approximation of function |x|s. Journal of the Belarusian State University. Mathematics and Informatics, 3, 18-34. https://doi.org/10.33581/2520-6508-2019-3-18-34
Issue
No 3 (2019): Journal of the Belarusian State University. Mathematics and Informatics
Section
Real, Complex and Functional Analysis

Copyright (c) 2019 Journal of the Belarusian State University. Mathematics and Informatics

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

The authors who are published in this journal agree to the following:

  1. The authors retain copyright on the work and provide the journal with the right of first publication of the work on condition of license Creative Commons Attribution-NonCommercial. 4.0 International (CC BY-NC 4.0).
  2. The authors retain the right to enter into certain contractual agreements relating to the non-exclusive distribution of the published version of the work (e.g. post it on the institutional repository, publication in the book), with the reference to its original publication in this journal.
  3. The authors have the right to post their work on the Internet (e.g. on the institutional store or personal website) prior to and during the review process, conducted by the journal, as this may lead to a productive discussion and a large number of references to this work. (See The Effect of Open Access.)