Skip to main content
SpringerLink
Log in

Direct and Converse Results on Weighted Simultaneous Approximation by the Method of Operators of Baskakov-Durrmeyer Type

  • Published:
Results in Mathematics Aims and scope Submit manuscript

Abstract

We consider the n- th. operators of Baskakov-Durrmeyer type, which result from the classical Baskakov-type operators with weights p nk, if the values f(k/n) are replaced by the integral terms \((n - c)\int_{I} p_{nk} (t) f(t) dt \), where I denotes the corresponding interval which depends on the parameter c. For integrable functions the rate of weighted simultaneous approximation will be related to the Ditzian-Totik modulus of smoothness.

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. V. A. Baskakov, An example of a sequence of linear positive operators in the space of continuous functions, Dokl. Akad. Nauk SSSR, 113 (1957), 249–251.

    MathSciNet  MATH  Google Scholar 

  2. M. Becker, R. J. Nessel, An elementary approach to inverse approximation theorems, J. Approx. Theory, 23 (1978), 99–103.

    Article  MathSciNet  MATH  Google Scholar 

  3. H. Berens, G. G. Lorentz, Inverse theorems for Bernstein polynomials, Indiana Univ. Math. J., 21 (1972), 693–708.

    Article  MathSciNet  MATH  Google Scholar 

  4. J. Bergh, J. LÖfström, Interpolation spaces, An Introduction, Springer-Verlag, Berlin, Heidelberg, New York, 1976.

    Book  MATH  Google Scholar 

  5. Z. Ditzian, K. Ivanov, Bernstein-type operators and their derivatives, J. Approx. Theory, 56 (1989), 72–90.

    Article  MathSciNet  MATH  Google Scholar 

  6. Z. Dltzian and V. Totik, Moduli of smoothness, Springer Series in Computational Mathematics 9, Springer-Verlag, Berlin, Heidelberg, New York, 1987.

    Google Scholar 

  7. J. L. DuRrmeyer, Une formule d'inversion de la transformée de Laplace: applications à la théorie des moments, Thèse de 3e cycle, Faculté des Sciences de l'Université de Paris, 1967.

  8. M. Heilmann, Commutativity of operators of Baskakov-Durrmeyer type, in: Constructive Theory of Functions '87 (Proc. Int. Conf. on Constr. Theory of Functions, Varna, May 24–31, 1987), Publ. House of the Bulgarian Academy of Sciences, Sofia, 1988, 197–206.

  9. M. Heilmann, Approximation auf [0,∞o) durch das Verfahren der Operatoren vom Baskakov-Durrrneyer Typ, Dissertation, Universität Dortmund, 1987.

  10. M. Heilmann, Direct and converse results for operators of Baskakov-Durrmeyer type, Approx. Theory and its Appl., 5 (1989), 105–127.

    MathSciNet  MATH  Google Scholar 

  11. M. Heilmann, On simultaneous approximation by optimal algebraic polynomials, to appear in Results in Mathematics.

  12. M. Heilmann and M. W. Müller, On simultaneous approximation by the method of Baskakov-Durrmeyer operators, Numer. Funct. Anal, and Optimiz., 10 (1989), 127–138.

    Article  MATH  Google Scholar 

  13. M. Heilmann and M. W. MÜller, Direct and converse results on simultaneous approximation by the method of Bernstein-Durrmeyer operators, to appear in Proc. Conf. on Algorithms for Approximation, RMCS Shrivenham, July 12–15, 1988.

  14. M. Heilmann and M. W. Müller, Equivalence of a weighted modulus of smoothness and a modified weighted K-functional, submitted to J. Approx. Theory.

  15. S. M. Mazhar, V. Totik, Approximation by modified Szász operators, Ada Sci. Math., 49 (1985), 257–269.

    MathSciNet  MATH  Google Scholar 

  16. F. Schurer, On linear positive operators in approximation theory, Ph. D. Thesis, Technological University of Delft, 1965.

Download references

Author information

Authors and Affiliations

  1. Lehrstuhl Mathematik VIII, Universität Dortmund, Postfach 50 05 00, D-4600, Dortmund 50, Fed. Rep. of Germany

    Margareta Heilmann & Manfred W. Müller

Authors
  1. Margareta Heilmann
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Manfred W. Müller
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Dedicated to Prof. Dr. Karl Zeller on the occasion of his 65th birthday

Rights and permissions

Reprints and permissions

About this article

Cite this article

Heilmann, M., Müller, M.W. Direct and Converse Results on Weighted Simultaneous Approximation by the Method of Operators of Baskakov-Durrmeyer Type. Results. Math. 16, 228–242 (1989). https://doi.org/10.1007/BF03322474

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03322474

Keywords

Search

Navigation