Skip to main content
SpringerLink
Log in

Jackson type theorems on complex curves

  • Published:
Science in China Series A: Mathematics Aims and scope Submit manuscript

Abstract

Let Γ be a closed smooth Jordan curve in the complex plane. In this paper, with the help of a class of fundamental functions of Hermite interpolation, the author introduces a continuous function interpolation which uniformly approximates to f(z) ε C(Γ) with the same order of approximation as that in Jackson Theorem 1 on real interval [−1, 1]. The accuracy of the order of approximation is proved. Using the method different from the early works, the author studies simultaneous approximation to function and its derivatives and the desired results analogues to that in Jackson Theorem 2 on real interval [−1, 1] are obtained.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Newman D J. Jackson’s theorem on complex arcs. J Approx Theory, 10: 206–217 (1974)

    Article  MATH  Google Scholar 

  2. Lesley F D. Best approximation on arcs. J Approx Theory, 18: 378–382 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  3. Andrievskii V. Harmonic version of Jackson’s theorem in the complex plane. J Approx Theory, 90: 224–234 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  4. Mergelyan S N. On the representation of functions by series of polynomials on closed sets (in Russian). Dokl Akad Nauk SSSR, 78: 405–408 (1951)

    MathSciNet  Google Scholar 

  5. Dzyadyk V K. Introduction of Uniform Approximation to Functions by Polynomials (in Russian). Moscow: Nauk, 1977, 339, 383–384

    MATH  Google Scholar 

  6. Walsh J L. Interpolation and Approximation by Rational Functions in the Complex Domain. Providence: AMS, 1956, 357

    Google Scholar 

  7. Wang Z Y, Zhu C Q. The order of approximation by complex rational type interpolation operators at Fejér points (in Chinese). Chinese Quart J Math, 7(3): 68–70 (1992)

    MathSciNet  Google Scholar 

  8. Al’per S Y. On uniform approximation to functions of a complex variable in closed domain (in Russian). Izv Akad Nauk SSSR Ser Math, 19: 423–444 (1955)

    MATH  MathSciNet  Google Scholar 

  9. Al’per S Y. On convergence of Lagrange’s interpolation polynomials in complex domain (in Russian). Uspekhi Mat Nauk, 11(5): 44–50 (1956)

    MATH  MathSciNet  Google Scholar 

  10. Tu T L. The approximation order of a kind of complex rational type interpolation (in Chinese). J North China University of Hydroeletricity, 23(3): 73–77 (2002)

    Google Scholar 

  11. Tu T L. Complex type interpolation uniformly approximates a function and its derivative. J Math (Wuhan), 26(4): 379–388 (2006)

    MATH  MathSciNet  Google Scholar 

  12. Shen X C, Tu T L. Complex interpolatary approximation. Adv Math (China), 20(2): 152–179 (1991)

    MATH  MathSciNet  Google Scholar 

  13. Natanson N P. Constructive Theory of Functions (in Russian). Moscow-Leningrad: GIT-TL, 1949, 117, 121

    Google Scholar 

  14. Fejér L. Ueber interpolation. Nachr Akad Wiss G"ottinger, 15: 66–91 (1916)

    Google Scholar 

  15. Szabados J. On some interpolatory procedures based on the roots of unity. Acta Math Hungar, 25(1–2): 159–164 (1974)

    MATH  Google Scholar 

  16. Tu T L. The Simultaneous approximation order by Hermite interpolation in a smooth domain. Acta Math Sin (Engl Ser), 18(4): 631–646 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  17. Tu T L. The Bernstein type inequality and simultaneous approximation by interpolation polynomials in complex domain. Acta Math Sci Ser B Engl Ed, 11(1): 213–220 (1991)

    MATH  Google Scholar 

  18. Lee K P, Liu H J, Qiu H J. Theory of Modern Function (in Chinese). Wuhan: Wuhan University Press, 1983, 242–244

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Deptartment of Mathematics and Informatics, North China Institute of Water Conservancy and Hydroelectric Power, Zhengzhou, 450011, China

    TianLiang Tu

Authors
  1. TianLiang Tu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to TianLiang Tu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tu, T. Jackson type theorems on complex curves. Sci. China Ser. A-Math. 52, 493–506 (2009). https://doi.org/10.1007/s11425-008-0144-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-008-0144-6

Keywords

MSC(2000)

Search

Navigation