Abstract
It is known that Szász–Durrmeyer operator is the limit, in an appropriate sense, of Bernstein–Durrmeyer operators. In this paper, we adopt a new technique that comes from the representation of operator semigroups to study the approximation issue as mentioned above. We provide some new results on approximating Szász–Durrmeyer operator by Bernstein–Durrmeyer operators. Our results improve the corresponding results of Adell and De La Cal (Comput Math Appl 30:1–14, 1995).
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Abel U., Berdysheva E.E.: Complete asymptotic expansion for multivariate Bernstein–Durrmeyer operators and quasi-interpolants. J. Approx. Theory 162, 201–220 (2010)
Adell J.A., De La Cal J.: Bernstein–Durrmeyer Operators. Comput. Math. Appl. 30, 1–14 (1995)
Batir N.: Sharp inequalities for factorial n. Proyecciones 27, 97–102 (2008)
Butzer P.L., Berens H.: Semi-Groups of Operators and Approximation. Springer, Berlin (1967)
Cárdenas-Morales D., Garrancho P., Raşa I.: Approximation properties of Bernstein–Durrmeyer type operators. Appl. Math. Comput. 232, 1–8 (2014)
Cárdenas-Morales D., Gupta V.: Two families of Bernstein–Durrmeyer type operators. Appl. Math. Comput. 248, 342–353 (2014)
De La Cal J., Luquin F.: A note on limiting properties of some Bernstein-type operators. J. Approx. Theory 68, 322–329 (1992)
De La Cal J., Luquin F.: Approximating Szász and Gamma operators by Baskakov operators. J. Math. Anal. Appl. 184, 585–593 (1994)
Derriennic M.M.: Sur l‘approximation de functions integrable sur [0,1] par des polynomes de Bernstein modifiés. J. Approx. Theory 31, 325–343 (1981)
Durrmeyer J.L.: Thése de 3e cycle. Faculté des Sciences de l‘Université de Paris, Paris (1967)
Gomilko A., Tomilov Y.: On convergence rates in approximation theory for operator semigroups. J. Funct. Anal. 266, 3040–3082 (2014)
Haight F.A.: Handbook of the Poisson Distribution. Wiley, New York (1968)
Kacsó D.: The limiting semigroup of the genuine Bernstein–Durrmeyer iterates. Results Math. 53, 303–310 (2009)
Makabe H., Morimura H.: On the approximation to some limiting distributions. Kōdai Math. Sem. Rep. 8, 31–40 (1956)
Mazhar S.M., Totik V.: Approximation by modified Szász operators. Acta Sci. Math. (Szeged) 49, 257–269 (1985)
Pfeifer D.: Approximation-theoretic aspects of probabilistic representations for operator semigroups. J. Approx. Theory 43, 271–296 (1985)
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Xu, XW., Zeng, XM. Asymptotic Properties of Bernstein–Durrmeyer Operators. Results. Math. 69, 345–357 (2016). https://doi.org/10.1007/s00025-015-0482-y
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DOI: https://doi.org/10.1007/s00025-015-0482-y