Abstract
The present paper deals with the estimate of the differences of certain positive linear operators and their derivatives. Oxur approach involves operators defined on bounded intervals, as Bernstein operators, Kantorovich operators, genuine Bernstein-Durrmeyer operators, and Durrmeyer operators with Jacobi weights. The estimates in quantitative form are given in terms of the first modulus of continuity. In order to analyze the theoretical results in the last section, we consider some numerical examples.
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References
Abel, U., Heilmann, M.: The complete asymptotic expansion for Bernstein-Durrmeyer operators with jacobi weights. Mediterr. J. Math. 1, 487–499 (2004)
Acu, A.M., Rasa, I.: New estimates for the differences of positive linear operators. Numer. Algorithm. 73(3), 775–789 (2016)
Adell, J.A., Lekuona, A.: Binomial convolution and transformations of Appell polynomials. J. Math. Anal Appl. 456(1), 16–33 (2017)
Adell, J.A., Lekuona, A.: A probabilistic generalization of the Stirling numbers of the second kind. J. Number Theory 194, 335–355 (2019)
Adell, J.A., Lekuona, A.: Explicit expressions for a certain class of Apell polynomials. A probabilistic approach, submitted for publication, arXiv:http://arXiv.org/abs/1711.02603v1[math.NT]
Aral, A., Inoan, D., Raşa, I.: On differences of linear positive operators. Anal. Math. Phys. https://doi.org/10.1007/s13324-018-0227-7 (2018)
Berens, H., Xu, Y.: On Bernstein-Durrmeyer Polynomials with Jacobi Weights. In: Chui, C.K. (ed.) Approximation Theory and Functional Analysis, pp 25–46. Academic Press, Boston (1991)
Berens, H., Xu, Y.: On Bernstein-Durrmeyer polynomials with Jacobi weights: the cases p = 1 and p = 1. In: Baron, S., Leviatan, D. (eds.) Approximation, Interpolation and Summation (Israel Math. Conf. Proc., 4), pp 51–62. Bar-Ilan University, Ramat Gan (1991)
Bernstein, S.N.: Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités. Commun. Soc. Math Kharkov 13, 1–2 (1913)
Beutel, L., Gonska, H., Kacsó, D., Tachev, G.: Variation-diminishing splines revised. In: Trâmbiţaş, R. (ed.) Proc. Int. Sympos. on Numerical Analysis and Approximation Theory, pp 54–75. Presa Universitară Clujeană, Cluj-Napoca (2002)
Durrmeyer, J.L.: Une formule d’inversion de la transforme de Laplace: applications a la theorie des moments. These de 3e cycle, Paris (1967)
Gonska, H., Piţul, P., Raşa, I.: On differences of positive linear operators, Carpathian. J. Math. 22(1–2), 65–78 (2006)
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 (Proc. Int. Conf. Cluj-Napoca 2006; ed. by O. Agratini and P. Blaga), pp. 55–80. Cluj-Napoca, Casa Cărţii de Ştiinţă (2006)
Gonska, H., Raşa, I.: Differences of positive linear operators and the second order modulus. Carpathian J. Math. 24(3), 332–340 (2008)
Goodman, T.N.T., Sharma, A.: A modified Bernstein-Schoenberg operator, Proc. of the Conference on Constructive Theory of Functions, Varna 1987 (ed. by Bl. Sendov et al.). Sofia: Publ. House Bulg. Acad. of Sci., pp. 166–173 (1988)
Kantorovich, L.V.: Sur certains developpements suivant les polynômes de la forme de S. Bernstein I, II. Dokl. Akad. Nauk. SSSR 563-568, 595–600 (1930)
Lupaş, A.: Die Folge Der Betaoperatoren. Dissertation, Universität Stuttgart (1972)
Lupaş, A.: The approximation by means of some linear positive operators. In: Müller, M.W. et al. (eds.) Approximation Theory, pp 201–227. Akademie-Verlag, Berlin (1995)
Păltănea, R.: Sur un opérateur polynomial defini sur l’ensemble des fonctions intégrables. Babes-Bolyai Univ. Fac. Math. Comput. Sci. Res. Semin. 2, 101–106 (1983)
Raşa, I.: Discrete operators associated with certain integral operators. Stud. Univ. Babeş Bolyai Math. 56(2), 537–544 (2011)
Raşa, I., Stănilă, E.: On some operators linking the Bernstein and the genuine Bernstein-Durrmeyer operators. J. Appl. Funct. Anal. 9, 369–378 (2014)
Tenberg, R.: Linearkombinationen Von Bernstein-Durrmeyer-Polynomen Bzgl. Jacobi-Gewichtsfunktionen, Diplomarbeit, Univ, Dortmund, Fachbereich Mathematik (1994)
Acknowledgments
We are very grateful to the referees for their highly valuable comments and remarks. Special thanks are due for the suggestions leading to inequality (5) and to the actual form of Lemma 4.1.
Funding
The work of the first author was financed by the Lucian Blaga University of Sibiu and Hasso Plattner Foundation research grants LBUS-IRG-2019-05.
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Acu, AM., Raşa, I. Estimates for the differences of positive linear operators and their derivatives. Numer Algor 85, 191–208 (2020). https://doi.org/10.1007/s11075-019-00809-4
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DOI: https://doi.org/10.1007/s11075-019-00809-4
Keywords
- Positive linear operators
- Differences of operators
- Bernstein operators
- Durrmeyer operators
- Kantorovich operators