On a Lebesgue constant of interpolation rational process at the Chebyshev – Markov nodes

Authors

  • Yauheni A. Rouba Yanka Kupala State University of Grodno, 22 Ažeška Street, Grodno 230023, Belarus
  • Kanstantin A. Smatrytski Yanka Kupala State University of Grodno, 22 Ažeška Street, Grodno 230023, Belarus
  • Yauheni V. Dirvuk Belarusian State University, Niezaliežnasci Avenue, 4, 220030, Minsk, Belarus

Keywords:

rational approximation, interpolation, Chebyshev – Markov fraction, Lebesgue constant

Abstract

In the present paper estimate of a Lebesgue constant of the interpolation rational Lagrange process on the segment [−1 ,1]  at the Chebyshev – Markov cosine fractions nodes is considered. It is shown that in the case of two real geometrically distinct poles of approximating functions, the norms of the Lagrange fundamental polynomials are bounded. Based on this result, it is proved that in the case under consideration the upper estimate of the Lebesgue constant does not depend on the arrangement of the poles and the sequence of the Lebesgue constant grows with logarithmic rate. Note, that in previous works the estimates of Lebesgue constants were obtained only for particular choices of poles or depended on the arrangement of poles.

Author Biographies

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

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

  • Yauheni V. Dirvuk, Belarusian State University, Niezaliežnasci Avenue, 4, 220030, Minsk, Belarus

    PhD (physics and mathematics); associate professor at the department of system programming and computer security, faculty of mathematics and informatics

References

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  2. Dzyadyk VK. Vvedenie v teoriyu ravnomernogo priblizheniya funktsii polinomami [An Introduction to the theory of uniform approximation of functions by polynomials]. Moscow: Nauka; 1977. 511 p. Russian.
  3. Rusak VN. [On interpolation of rational functions with fixed poles]. Doklady Akademii nauk Belorusskoi SSR. 1962;4(9):548–550. Russian.
  4. Starovoitov AP. [On rational interpolation with fixed poles]. Izvestiya AN BSSR. Seriya fiziko-matematicheskikh nauk. 1983; 6:105–106. Russian.
  5. Rouba YA, Dirvuk YU. [On a Lebesgue constant of interpolation rational Lagrange processes at the Chebyshev – Markov nodes]. Izvestiya NAN Belarusi. Seriya fiziko-matematicheskikh nauk. 2015;4:32–38. Russian.
  6. Rouba YA, Kazlouskaya NY. [On a problem of estimation of a Lebesgue constant of interpolation rational processes at the Chebyshev – Markov nodes]. Vesnik Hrodzenskaha dziarzhaunaha universiteta imia Ianki Kupaly. Seriya 2. 2016;6(3):6 –11. Russian.
  7. Natanson IP. Konstruktivnaya teoriya funktsii [Constructive function theory]. Moscow, Leningrad: Gostekhizdat; 1949. 688 p. Russian.

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Published

2019-01-19

Issue

No. 3 (2018): Journal of the Belarusian State University. Mathematics and Informatics

Section

Real, Complex and Functional Analysis

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How to Cite

[1]
Rouba, Y.A. et al. 2019. On a Lebesgue constant of interpolation rational process at the Chebyshev – Markov nodes. Journal of the Belarusian State University. Mathematics and Informatics. 3 (Jan. 2019), 12–20.