Skip to main content
Log in

On the Composition and Decomposition of Positive Linear Operators (V)

  • Published:
Results in Mathematics Aims and scope Submit manuscript

Abstract

The present note supplements information contained in three earlier papers with the same title. Here we consider certain aspects of (in)decomposability of positive linear operators given on C(X) where X is a compact convex metrizable subset of a topological vector space which is locally convex and Hausdorff.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Aldaz, J.M., Render, H.: Optimality of generalized Bernstein operators. J. Approx. Theory 162, 1407–1416 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  2. Altomare, F., Campiti, M.: Korovkin-type Approximation Theory and its Applications. Walter de Gruyter, Berlin (1994)

    Book  MATH  Google Scholar 

  3. Altomare, F., Cappelletti Montano, M., Leonessa, V., Raşa, I.: Markov Operators, Positive Semigroups and Approximation Processes. Walter de Gruyter, Berlin (2014)

  4. Berens, H., DeVore, R.: A Characterization of Bernstein Polynomials. In: Approximation Theory III, pp. 213–219. Academic Press, New York (1980)

  5. Beśka, M.: Convexity and variation diminishing property of multidimensional Bernstein polynomials. Approx. Theory Appl. 5, 59–78 (1989)

    MATH  MathSciNet  Google Scholar 

  6. Bustamante, J., Quesada, J.M.: On an extremal relation of Bernstein operators. J. Approx. Theory 141, 214–215 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  7. Cárdenas-Morales, D., Garrancho, P., Raşa, I.: Optimality of piecewise \(\tau \)-linear interpolating operators. Appl. Math. Comput. 219, 6445–6448 (2013)

    MATH  MathSciNet  Google Scholar 

  8. Dahmen, W.: Convexity and Bernstein–Bézier polynomials. In: Curves and Surfaces, pp. 107–134. Academic Press (1991)

  9. Dahmen, W., Micchelli, C.A.: Convexity of multivariate Bernstein polynomials and box spline surfaces. Studia Sci. Math. Hungar. 23, 265–287 (1988)

    MATH  MathSciNet  Google Scholar 

  10. Dahmen, W., Micchelli, C.A.: Convexity and Bernstein polynomials on \(k\)-simploids. Acta Math. Appl. Sin. 6, 50–66 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  11. de la Cal, J., Cárcamo, J.: An extremal property of Bernstein operators. J. Approx. Theory 146, 87–90 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  12. Gavrea, I., Gonska, H., Kacsó, D.: Variation on Butzer’s problem: characterization of the solutions. Comput. Math. Appl. 34(9), 51–64 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  13. Gavrea, I., Ivan, M.: An extremal property for a class of positive linear operators. J. Approx. Theory 162, 6–9 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  14. Gonska, H.: On the composition and decomposition of positive linear operators. In: Approximation Theory and its Applications (Ukrainian), pp. 161–180. Pr. Inst. Mat. Nats. Akad. Nauk Ukr. Mat. Zastos, Kiev (1999)

  15. Gonska, H., Heilmann, M., Lupaş, A., Raşa, I.: On the composition and decomposition of positive linear operators (III): A non-trivial decomposition of the Bernstein operator (2012). arXiv:1204.2723

  16. Gonska, H., Piţul, P., Raşa, I.: On Peano’s form of the Taylor remainder, Voronovskaja’s theorem and the commutator of positive linear operators. In: Numerical Analysis and Approximation Theory, pp. 55–80. Casa Cărţii de Ştiinţa, Cluj-Napoca (2006)

  17. Gonska, H., Raşa, I.: On the composition and decomposition of positive linear operators (II). Studia Sci. Math. Hungar. 47, 448–461 (2010)

    MATH  MathSciNet  Google Scholar 

  18. Heilmann, M., Raşa, I.: On the decomposition of Bernstein operators. Numer. Funct. Anal. Optim. 36, 72–85 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  19. Heilmann, M., Nasaireh, F., Raşa, I.: Beta and related operators revisited (preprint) (2016)

  20. Lupaş, A.: The approximation by means of some linear positive operators. In: Approximation Theory (Witten, 1995), pp. 201–229. Akademie-Verlag, Berlin (1995)

  21. Micchelli, C.A.: Convergence of positive linear operators on \(C(X)\). J. Approx. Theory 13, 305–315 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  22. Raşa, I.: On some results of C.A. Micchelli. Anal. Numér. Théor. Approx. 9, 125–127 (1980)

  23. Raşa, I.: Sets on which concave functions are affine and Korovkin closures. Anal. Numér. Théor. Approx. 15, 163–165 (1986)

    MATH  MathSciNet  Google Scholar 

  24. Sauer, T.: Multivariate Bernstein polynomials and convexity. Comput. Aided Geom. Design 8, 465–478 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  25. Sauer, T.: Ein Bernstein–Durrmeyer-Operator auf dem Simplex, Dissertation, Universität Erlangen-Nürnberg (1992)

  26. Waldron, Sh: A generalized beta integral and the limit of the Bernstein–Durrmeyer operator with Jacobi weights. J. Approx. Theory 122, 141–150 (2003)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Heiner Gonska.

Additional information

Dedicated to our dear colleague Professor Margareta Heilmann on the occasion of her 60th birthday.

This work was completed with the support of our \({\hbox{\TeX}}\)-pert Birgit Dunkel. We thank her very much.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gonska, H., Raşa, I. On the Composition and Decomposition of Positive Linear Operators (V). Results Math 72, 1033–1040 (2017). https://doi.org/10.1007/s00025-016-0618-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00025-016-0618-8

Keywords

Mathematics Subject Classification

Navigation