Skip to main content
SpringerLink
Log in

Bernstein Type Asymptotic Evaluations for Linear Positive Operators on \(C\left[ \alpha ,\beta \right] \)

  • Published:
Results in Mathematics Aims and scope Submit manuscript

Abstract

We prove a general Bernstein type asymptotic evaluation for linear positive operators on the space \(C\left[ \alpha ,\beta \right] \). Our proof is entirely different than those known in the literature. As application, we prove a Bernstein type asymptotic evaluation for some general Bernstein kind of linear positive operators, which extend the classical Bernstein extension of Voronovskaja’s theorem. We use then these general results to get, for many concrete linear positive operators, the Voronovskaja–Bernstein type results.

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. Altomare, F., Campiti, M.: Korovkin-Type Approximation Theory and Its Applications. De Gruyter Studies in Mathematics, vol. 17. Walter de Gruyter & Co., Berlin (1994)

    Book  Google Scholar 

  2. Altomare, F., Montano, M.C., Leonessa, V., Raşa, I.: A generalization of Kantorovich operators for convex compact subsets. Banach J. Math. Anal. 11(3), 591–614 (2017)

    Article  MathSciNet  Google Scholar 

  3. Altomare, F., Montano, M.C., Leonessa, V., Raşa, I.: Elliptic differential operators and positive semigroups associated with generalized Kantorovich operators. J. Math. Anal. Appl. 458(1), 153–173 (2018)

    Article  MathSciNet  Google Scholar 

  4. Bernstein, S.: Complément ‘a l’article de E. Voronovskaya “Détermination de la forme asymptotique de l’approximation des fonctions par les polynômes de M. Bernstein”, C. R. Acad. Sc. URSS 1932 86–92 (1932)

  5. Cao, J.D.: A generalization of Bernstein polinomials. J. Math. Anal. Appl. 209(2), 140–146 (1997)

    Article  MathSciNet  Google Scholar 

  6. Davis, P.J.: Interpolation and Approximation, Dover Publications, Inc., New York, 1975. Republication, with minor corrections, of the 1963 original, with a new preface and bibliography

  7. de la Cal, J., Valle, A.M.: A generalization of Bernstein–Kantorovič operators. J. Math. Anal. Appl. 252(2), 750–766 (2000)

    Article  MathSciNet  Google Scholar 

  8. DeVore, R.A., Lorentz, G.G.: Constructive Approximation. Springer, Berlin (1993)

    Book  Google Scholar 

  9. Ding, C.: Approximation by generalised Bernstein polynomials. Indian J. Pure Appl. Math. 35(6), 817–826 (2004)

    MathSciNet  MATH  Google Scholar 

  10. Gavrea, I., Ivan, M.: The Bernstein Voronovskaja-type theorem for positive linear approximation operators. J. Approx. Theory 192, 291–296 (2015)

    Article  MathSciNet  Google Scholar 

  11. Gavrea, I., Ivan, M.: Complete asymptotic expansions related to conjecture on a Voronovskaja-type theorem. J. Math. Anal. Appl. 458(1), 452–463 (2018)

    Article  MathSciNet  Google Scholar 

  12. Korovkin, P.P.: On convergence of linear positive operators in the space of continuous functions. Doklady Akademii Nauk 90, 961–964 (1953). (Russian)

    MathSciNet  Google Scholar 

  13. Korovkin, P.P.: Linear Operators and Approximation Theory. Hindustan Publ. Corp, New Delhi (1960)

    Google Scholar 

  14. Lorentz, G.G.: Bernstein Polynomials. AMS Chelsea Publishing, New York (1997)

    MATH  Google Scholar 

  15. Nagel, J.: Asymptotic properties of powers of Bernstein operators. J. Approx. Theory 29, 323–335 (1980)

    Article  MathSciNet  Google Scholar 

  16. Nagel, J.: Asymptotic properties of powers of Kantorovič operators. J. Approx. Theory 36, 268–275 (1982)

    Article  MathSciNet  Google Scholar 

  17. Popa, D.: Asymptotic evaluations for some sequences of positive linear operators. Constr. Approx. (2018). https://doi.org/10.1007/s00365-018-9449-z

    Article  Google Scholar 

  18. Sikkema, P.C.: On some linear positive operators. Nederl. Akad. Wet., Proc. Ser. A 73, 327–337 (1970)

    MathSciNet  MATH  Google Scholar 

  19. Tachev, G.T.: Voronovskaja’s theorem revisited. J. Math. Anal. Appl. 343(1), 399–404 (2008)

    Article  MathSciNet  Google Scholar 

  20. Tachev, G.T.: The complete asymptotic expansion for Bernstein operators. J. Math. Anal. Appl. 385(2), 1179–1183 (2012)

    Article  MathSciNet  Google Scholar 

  21. Voronovskaja, E.: Determination de la forme asymptotique d’approximation des fonctions par les polynômes de M. Bernstein. Dokl. Akad. Nauk SSSR 4, 86–92 (1932)

    MATH  Google Scholar 

Download references

Acknowledgements

We would like to thank the two referees of our paper for carefully reading the manuscript and for such constructive comments, remarks and suggestions which helped improving the first version of the paper.

Author information

Authors and Affiliations

  1. Department of Mathematics, Ovidius University of Constanta, Bd. Mamaia 124, 900527, Constanţa, Romania

    Dumitru Popa

Authors
  1. Dumitru Popa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dumitru Popa.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Popa, D. Bernstein Type Asymptotic Evaluations for Linear Positive Operators on \(C\left[ \alpha ,\beta \right] \). Results Math 74, 177 (2019). https://doi.org/10.1007/s00025-019-1101-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00025-019-1101-0

Keywords

Mathematics Subject Classification

Search

Navigation