Advertisement

Mathematical Notes

, Volume 104, Issue 5–6, pp 848–858 | Cite as

Estimates of the Best Approximation of Polynomials by Simple Partial Fractions

  • M. A. KomarovEmail author
Article
  • 4 Downloads

Abstract

An asymptotics of the error of interpolation of real constants at Chebyshev nodes is obtained. Some well-known estimates of the best approximation by simple partial fractions (logarithmic derivatives of algebraic polynomials) of real constants in the closed interval [−1, 1] and complex constants in the unit disk are refined. As a consequence, new estimates of the best approximation of real polynomials on closed intervals of the real axis and of complex polynomials on arbitrary compact sets are obtained.

Keywords

simple partial fraction approximation estimate best approximation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    V. I. Danchenko and D. Ya. Danchenko, “On uniform approximation of polynomials by logarithmic derivatives,” in Function Theory, Its Applications, and Related Issues (Kazan, 1999), pp. 74–77 [in Russian].Google Scholar
  2. 2.
    V. I. Danchenko and D. Ya. Danchenko, “Approximation by simplest fractions,” Mat. Zametki 70 (4), 553–559 (2001) [Math. Notes 70 (4), 502–507 (2001)].MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    O. N. Kosukhin, “Approximation properties of the most simple fractions,” Vestnik Moskov. Univ. Ser. I Mat. Mekh., No. 4, 54–59 (2001) [Moscow Univ.Math. Bull. 56 (4), 36–40 (2001)].MathSciNetzbMATHGoogle Scholar
  4. 4.
    O. N. Kosukhin, On Nontraditional Methods of Approximation Related to Complex Polynomials, Cand. Sci. (Phys.–Math.) Dissertation (Moskov. Univ., Moscow, 2005) [in Russian].Google Scholar
  5. 5.
    E. N. Kondakova, “Interpolation by the simplest fractions,” Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform. 9 (2), 30–37 (2009).Google Scholar
  6. 6.
    M. A. Komarov, “A criterion for the best approximation of constants by simple partial fractions,” Mat. Zametki 93 (2), 209–215 (2013) [Math. Notes 93 (2), 250–256 (2013)].MathSciNetCrossRefGoogle Scholar
  7. 7.
    M. A. Komarov, “Best approximation rate of constants by simple partial fractions and Chebyshev alternance,” Mat. Zametki 97 (5), 718–732 (2015) [Math. Notes 97 (5), 725–737 (2015)].MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    V. I. Danchenko, “Estimates of derivatives of simplest fractions and other questions,” Mat. Sb. 197 (4), 33–52 (2006) [Sb.Math. 197 (4), 505–524 (2006)].MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    V. I. Danchenko, “Approximation properties of sums of the form Σkλk hk z),” Mat. Zametki 83 (5), 643–649 (2008) [Math. Notes 83 (5), 587–593 (2008)].MathSciNetCrossRefGoogle Scholar
  10. 10.
    P. Chunaev and V. Danchenko, “Approximation by amplitude and frequency operators,” J. Approx. Theory 207, 1–31 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    V. I. Danchenko and P. V. Chunaev, “Approximation by simple partial fractions and their generalizations,” J.Math. Sci. 176 (6), 844–859 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    M. A. Komarov, “A criterion for the best uniform approximation by simple partial fractions in terms of alternance. II,” Izv. Ross. Akad. Nauk Ser.Mat. 81 (3), 109–133 (2017) [Izv.Math. 81 (3), 568–591 (2017)].MathSciNetzbMATHGoogle Scholar
  13. 13.
    S. N. Bernstein, Extremal Properties of Polynomials and Best Approximation of Continuous Functions of a Real Variable (GONTI, Leningrad–Moscow, 1937) [in Russian].Google Scholar
  14. 14.
    M. A. Komarov, “On best approximation of real analytic functions by simple partial fractions,” in Abstracts of Papers Of the International Conference on Mathematical Control Theory and Mechanics, Suzdal, July 7–11, 2017 (OOO “Arkaim,” 2017), pp. 84–85 [in Russian].Google Scholar
  15. 15.
    M. A. Komarov, “A criterion for the best uniform approximation by simple partial fractions in terms of alternance,” Izv. Ross. Akad. Nauk Ser.Mat. 79 (3), 3–22 (2015) [Izv.Math. 79 (3), 431–448 (2015)].MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    V. K. Dzyadyk, “Asymptotics of diagonal Padéapproximants of the functions sin z, cos z, sinh z, and cosh z,” Mat. Sb. 108 (150) (2), 247–267 (1979) [Sb.Math. 36 (2), 231–249 (1980)].MathSciNetGoogle Scholar
  17. 17.
    H. Bateman and A. Erdélyi, Higher Transcendental Functions, Vol. 1: The Hypergeometric Function, Legendre Functions (McGraw–Hill, New York–Toronto–London, 1953; Nauka, Moscow, 1973).zbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Vladimir State UniversityVladimirRussia

Personalised recommendations