Abstract
In the present paper, we introduce the Bezier-variant of Durrmeyer modification of the Bernstein operators based on a function \(\tau \), which is infinite times continuously differentiable and strictly increasing function on [0, 1] such that \(\tau (0)=0\) and \(\tau (1)=1\). We give the rate of approximation of these operators in terms of usual modulus of continuity and K-functional. Next, we establish the quantitative Voronovskaja type theorem. In the last section we obtain the rate of convergence for functions having derivative of bounded variation.
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References
Acar, T., Aral, A., Rasa, I.: Modified Bernstein–Durrmeyer operators. Gen. Math. 22, 27–41 (2014)
Acar, T.: Asymptotic formulas for generalized Szasz–Mirakyan operators. Appl. Math. Comput. 263, 223–239 (2015)
Acar, T., Aral, A., Raşa, I.: The new forms of Voronovskaya’s theorem in weighted spaces. Positivity 20, 25–40 (2016)
Acu, A.M., Agrawal, P.N., Neer, T.: Approximation properties of the modified Stancu operators. Numer. Funct. Anal. Optim. doi:10.1080/01630563.20161248564
Agrawal, P.N., Ispir, N., Kajla, A.: Approximation properties of Bezier-summation-integral type operators based on Polya–Bernstein functions. Appl. Math. Comput. 259, 533–539 (2015)
Bernstein, S.N.: Démonstration du théorém de Weierstrass fondée sur la calcul des probabilitiés. Commun. Soc. math. Charkow Sér.2 t 13, 1–2 (1912)
Chang, G.: Generalized Bernstein–Bezier polynomials. J. Comput. Math. 1, 322–327 (1983)
Ditzian, Z., Totik, V.: Moduli of smoothness. Springer, New York (1987)
Finta, Z.: Remark on Voronovskaja theorem for \(q\)-Bernstein operators. Stud. Univ. Babeş Bolyai Math. 56, 335–339 (2011)
Gonska, H., Piţul, P., Raşa, I.: General King-type operators. Results Math. 53, 279–286 (2009)
Guo, S.S., Liu, G.F., Song, Z.J.: Approximation by Bernstein–Durrmeyer–Bezier operators in \(L_{p}\) spaces. Acta Math. Sci. Ser. A Chin. Ed. 30, 1424–1434 (2010)
King, J.P.: Positive linear operators which preserve \(x^{2}\). Acta Math. Hung. 99, 203–208 (2003)
Morales, D.C., Garrancho, P., Raşa, I.: Bernstein-type operators which preserve polynomials. Comput. Math. Appl. 62, 158–163 (2011)
Wang, P., Zhou, Y.: A new estimate on the rate of convergence of Durrmeyer–Bezier operators. J. Inequal. Appl. 2009, 702680 (2009). doi:10.1155/2009/702680
Zeng, X.M.: On the rate of convergence of two Bernstein–Bezier type operators for bounded variation functions II. J. Approx. Theory 104, 330–344 (2000)
Zeng, X.M., Piriou, A.: On the rate of convergence of two Bernstein–Bezier type operators for bounded variation functions. J. Approx. Theory 95, 369–387 (1998)
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Acar, T., Agrawal, P.N. & Neer, T. Bezier variant of the Bernstein–Durrmeyer type operators. Results Math 72, 1341–1358 (2017). https://doi.org/10.1007/s00025-016-0639-3
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DOI: https://doi.org/10.1007/s00025-016-0639-3