Approximation by the new modification of Bernstein-Stancu operators
Abstract
The current paper deals with the new modification of Bernstein-Stancu operators which preserve constant and Korovkin’s other test functions in limit case. We study the uniform convergence of the newly defined operators. The rate of convergence is investigated by means of the modulus of continuity, by using functions of Lipschitz class and by the help of Peetre-K functionals. Then a Voronovskaya type asymptotic formula for the newly constructed Bernstein-Stancu operators is presented. Finally, some graphs are given to illustrate the convergence properties of operators to some functions.
Keywords
Modulus of continuity, Peetre-K functionals, Voronovskaya-type theorem
References
- [1] Bernstein S.N., Démonstration du théorem de Weierstrass fondée sur le calculu des probabilités, Commun. Kharkov Math. Soc., 13(2) (1912) 1-2.
- [2] Stancu D.D., Approximation of function by means of a new generalized Bernstein operator, Calcolo, 20(2) (1983) 211-229.
- [3] Cai Q.-B., Lian B.-Y., Zhou G., Approximation properties of λ-Bernstein operators, J. Inequal. Appl., 61 (2018) 1-11.
- [4] Kajla A., Acar T., Blending type approximation by generalized Bernstein-Durrmeyer type operators, Miskolc Math. Notes, 19(1) (2018) 319-336.
- [5] Mohiuddine S.A., Özger F., Approximation of functions by Stancu variant of Bernstein-Kantorovich operators based on shape parameter α, RACSAM, 114(70) (2020) 1-17.
- [6] Srivastava H.M., Özger F., Mohiuddine SA., Construction of Stancu-type Bernstein operators based on Bezier bases with shape parameter λ, Symmetry, 11(3) (2019) 1-22.
- [7] Cai Q.-B., Dinlemez Kantar Ü., Çekim B., Approximation properties for the genuine modified Bernstein-Durrmeyer-Stancu operators, Appl. Math. J. Chinese Univ., 35(4) (2020) 468-478.
- [8] Cai Q.-B., Cheng W.-T., Çekim B., Bivariatea α, q-Bernstein-Kantorovich operators and GBS operators of bivariate α, q-Bernstein-Kantorovich type, Mathematics, 7(12) (2019) 1-18.
- [9] Usta F., On New Modification of Bernstein Operators: Theory and Applications, Iran J. Sci. Technol. Trans. Sci., 44 (2020) 1119–1124.
- [10] Korovkin P.P., On convergence of linear operators in the space of continuous functions (Russian), Dokl. Akad. Nauk SSSR (N.S.), 90 (1953) 961-964.