Statistical Approximation of the q-Bernstein-Durrmeyer Type Operators
In 2012, a kind of q-Bernstein-Durrmeyer type operators is introduced, and some approximate properties of these operators are studied by Ren. In this paper the statistical approximation properties of these operators are investigated. The Korovkin type statistical convergence theorem of these operators is established. Then the rates of statistical convergence of these operators are also studied by means of modulus of continuity and the help of functions of the Lipschitz class.
Keywordsq-Bernstein-Durrmeyer type operators q-integers Korovkin type theorem Rate of statistical convergence Modulus of continuity
This work is supported by the National Natural Science Foundation of China (No. 61170324), the Class A Science and Technology Project of Education Department of Fujian Province of China (No. JA12324), and the Natural Science Foundation of Fujian Province of China (No. 2013J01017 and No. 2014J01021).
- 5.Doǧru, O., Gupta, V.: Korovkin-type approximation properties of bivariate q-Meyer-König and Zeller operators. Calcolo 43(1), 51–63 (2012)Google Scholar
- 11.Mahmudov, N, Sabancigil, P.: A q-analogue of the Meyer-König and Zeller operators. Bull. Malays. Math. Sci. Soc. (2), 35(1), 39–51 (2012)Google Scholar
- 13.Ren, M Y: Approximation properties of the q-Bernstein-Durrmeyer type operators. Fuzzy Syst. Math. 26(5), 107–112 (2012) (Chinese)Google Scholar
- 14.Kac, V.G., Cheung, P.: Quantum Calculus. Universitext. Springer, New York (2002)Google Scholar
- 15.Gasper, G, Rahman, M.: Basic Hypergeometric Series. Encyclopedia of Mathematics and its Applications, vol. 35. Cambridge University Press, Cambridge (1990)Google Scholar
- 17.Niven, I., Zuckerman, H.S., Montgomery, H.: An Introduction to the Theory of Numbers. Wiley, New york (1991)Google Scholar
- 18.Doǧru O.: On statistical approximation properties of Stancu type bivariate generalization of \(q\)-Balás-Szabados operators. In: Seminar on Numerical Analysis and Approximation Theory, Cluj-Napoca, Univ. Babeş-Bolya, pp. 179–194 (2006)Google Scholar