Approximation properties of Bernstein-Kantorovich type operators of two variables

Year 2019, Volume: 68 Issue: 2, 2313 - 2323, 01.08.2019

Döne Karahan , Aydın İzgi

https://doi.org/10.31801/cfsuasmas.558169

Cited By: 1

Abstract

In this study, the generalized Bernstein-Kantorovich type operators

are introduced and some approximation properties of these operators

are studied in the space of continuous functions of two variables on

a compact set . The convergence rate of these operators are obtained by

means of the modulus of continuity. The Voronovskaya type theorem is

given and some differential properties of these operators are proved.

Keywords

Bernstein-Kantorovich operators, modulus of continuity, Voronovskaya type theorem

References

• Bernstein, S. N., Demonstration du theorem de Weierstrass fondee sur le calculu des probabilites,Comp. Comm. Soc. Mat. Charkow Ser., 13(2)(1912), 1-2.
• Korovkin, P. P., On convergence of linear positive operators in the space of continuousfunctions, Dokl. Akad. Nauk, 90(1953), 961-964.
• Kantorovich, L. V., Sur certains developments suivant les polynomes de la forms de S.Bernstein I, II, Dokal Akad Nauk SSSR, (1930) 595-600, 563-568.
• Durrmeyer, J. L., Une formula d’invension de la transforms de Laplace-Appliction a’la theorie des moments, The’se de 3e cycle, Faculte’ des Sciences de I’Universite de Paris, (1967).
• Izgi, A., Approximation by a class of new type Bernstein polynomials of one two variables,Global Journal of Pure and Applied Mathematics, 8(5) (2012), 55-71.
• Cao, J. D., A generalization of the Bernstein Polynomials, J. Math. Analy. and Appl.Math., 122(2000) (1997), 1-21.
• Lorentz, G. G., Bernstein polynomials, Chelsea, New York, (1986).
• Gurdek, M., Rempulska, L. and Skorupka, M., The Baskakov operators for functions oftwo variables, Collect. Math., 50(3) (1999), 289–302.
• Kahvecibasi, I., Approximation properties of the Bernstein-Kantorovich operators on theinterval [-1,1], Master of Science Thesis, Graduate School of Natural and Applied SciencesDepartment of Mathematic, Harran University, Sanlıurfa, Turkey.
• Volkov, V. I., On the convergence of sequences of linear positive operators in the space oftwo variables, Dokl. Akad. Nauk. SSSR (N.S.), 115 (1957), 17-19.
• Dirik, F. and Demirci, K., Korovkin type approximation theorem for functions of twovariables in statistical sense, Turk. J. Math., 34 (2010), 73–83.
• Stancu, D. D., A method for obtaining polynomials of Bernstein type of two variables,Amer. Math. Monthly, 70(3) (1963), 260-264.
• Gazanfer, A. K. and Büyükyazici, I., Approximation by certain linear positive operatorsof two variables, Hindawi Publishing Corporation Abstract and Applied Analysis, ID 782080,(2014).
• Sahai, A., An iterative reduced-bias algorithm for a dual-fusion variant of Bernstein’soperator, Inter. Journal of Math. Arch., 2(3) (2011), 331-334.