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.
Similar content being viewed by others
References
Aldaz, J.M., Render, H.: Optimality of generalized Bernstein operators. J. Approx. Theory 162, 1407–1416 (2010)
Altomare, F., Campiti, M.: Korovkin-type Approximation Theory and its Applications. Walter de Gruyter, Berlin (1994)
Altomare, F., Cappelletti Montano, M., Leonessa, V., Raşa, I.: Markov Operators, Positive Semigroups and Approximation Processes. Walter de Gruyter, Berlin (2014)
Berens, H., DeVore, R.: A Characterization of Bernstein Polynomials. In: Approximation Theory III, pp. 213–219. Academic Press, New York (1980)
Beśka, M.: Convexity and variation diminishing property of multidimensional Bernstein polynomials. Approx. Theory Appl. 5, 59–78 (1989)
Bustamante, J., Quesada, J.M.: On an extremal relation of Bernstein operators. J. Approx. Theory 141, 214–215 (2006)
Cárdenas-Morales, D., Garrancho, P., Raşa, I.: Optimality of piecewise \(\tau \)-linear interpolating operators. Appl. Math. Comput. 219, 6445–6448 (2013)
Dahmen, W.: Convexity and Bernstein–Bézier polynomials. In: Curves and Surfaces, pp. 107–134. Academic Press (1991)
Dahmen, W., Micchelli, C.A.: Convexity of multivariate Bernstein polynomials and box spline surfaces. Studia Sci. Math. Hungar. 23, 265–287 (1988)
Dahmen, W., Micchelli, C.A.: Convexity and Bernstein polynomials on \(k\)-simploids. Acta Math. Appl. Sin. 6, 50–66 (1990)
de la Cal, J., Cárcamo, J.: An extremal property of Bernstein operators. J. Approx. Theory 146, 87–90 (2007)
Gavrea, I., Gonska, H., Kacsó, D.: Variation on Butzer’s problem: characterization of the solutions. Comput. Math. Appl. 34(9), 51–64 (1997)
Gavrea, I., Ivan, M.: An extremal property for a class of positive linear operators. J. Approx. Theory 162, 6–9 (2010)
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)
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
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)
Gonska, H., Raşa, I.: On the composition and decomposition of positive linear operators (II). Studia Sci. Math. Hungar. 47, 448–461 (2010)
Heilmann, M., Raşa, I.: On the decomposition of Bernstein operators. Numer. Funct. Anal. Optim. 36, 72–85 (2015)
Heilmann, M., Nasaireh, F., Raşa, I.: Beta and related operators revisited (preprint) (2016)
Lupaş, A.: The approximation by means of some linear positive operators. In: Approximation Theory (Witten, 1995), pp. 201–229. Akademie-Verlag, Berlin (1995)
Micchelli, C.A.: Convergence of positive linear operators on \(C(X)\). J. Approx. Theory 13, 305–315 (1975)
Raşa, I.: On some results of C.A. Micchelli. Anal. Numér. Théor. Approx. 9, 125–127 (1980)
Raşa, I.: Sets on which concave functions are affine and Korovkin closures. Anal. Numér. Théor. Approx. 15, 163–165 (1986)
Sauer, T.: Multivariate Bernstein polynomials and convexity. Comput. Aided Geom. Design 8, 465–478 (1991)
Sauer, T.: Ein Bernstein–Durrmeyer-Operator auf dem Simplex, Dissertation, Universität Erlangen-Nürnberg (1992)
Waldron, Sh: A generalized beta integral and the limit of the Bernstein–Durrmeyer operator with Jacobi weights. J. Approx. Theory 122, 141–150 (2003)
Author information
Authors and Affiliations
Corresponding author
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
About this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00025-016-0618-8