Approximation Properties of Kantorovich Type Modifications of $(p, q)-$Meyer-König-Zeller Operators

Abstract

In this paper, we introduce Kantorovich type modification of $(p, q)$-Meyer-König-Zeller operators. We estimate rate of convergence of proposed operators using modulus of continuity and Lipschitz class functions. Further, we obtain the statistical convergence and local approximation results for these operators. In the last section, we estimate the rate of convergence of $(p, q)$-Meyer-König-Zeller Kantorovich operators by means of Matlab programming.

Keywords

• Modulus of continuity,Statistical Convergence,(p q)-Calculus,(p q)-Meyer-König-Zeller operators,Peetre's K-functional

References

1. [1] T. Acar: $(p, q)-$Genralization of S´zasz-Mirakyan operators. Math. Methods Appl. Sci., 39(10) (2016), 2685-2695.
2. [2] R. Chakrabarti and R. JAGANNATHAN: A $(p, q)-$oscillator realization of two parameter quantum algebras. J. Phys. A: Math. Gen., 24 (1991), 711-718.
3. [3] E. W. Cheney and A. Sharma: Bernstein power series. Canad. J. Math., 16 (1964),241-252.
4. [4] R. A. DeVore and G. G. Lorentz. Construtive Approximation, Springer, Berlin, (1993).
5. [5] O. Dogru and O. Duman: Statistical approximationof Meyer-König and Zeller operators based on $q-$integers. Publ. Math. Debrecen, 68(1-2) (2006), 199-214.
6. [6] O. Dogru, O. Duman and C. Orhan: Statistical approximation by generalized Meyer-König and Zeller type operators., Studia Sci. Math. Hungar, 40(3) (2003), 359-371.
7. [7] O. Dogru and V. Gupta: Kovrokin-type approximation properties of bivariate $q-$Meyer-König and Zeller operators. Calcolo, 43(1) (2006), 51-63.
8. [8] V. Gupta and A. Aral: Bernstein Durmeyer operators based on Two Parameters. SER. Math. Inform, 31(1) (2016), 79-95.

9. [9] V. Gupta and H. Sharma: Statistical approximation by $q-$integrated Meyer-König-Zeller-Kantorovich operators. Craetive Math. Inf., 19(1) (2010), 45-52.
10. [10] V. Gupta, H. Sharma, T. Kim and S. H. Lee: Properties of $q-$analogue of Beta operator. Advances in Difference Equations, 2012:86 (2012).
11. [11] U. Kadak, A. Khan and M. Mursaleen: Approximation by Meyer-König and Zeller Operators using $(p,q)-$Calculus. arXiv:1603.08539v2, (2016).
12. [12] K. Khan and D. K. Lobiyal: Bezier curves based on Lupas $(p,q)$ analogue of Bernstein polynomials in CAGD. arXiv: 1505.01810[cs.GR].
13. [13] P. P. Korovkin: Linear operators and approximation theory. Hindustan Publishing Corporation, Delhi, (1960).
14. [14] G. G. Lorentz: Berstein polynomials. Mathematical Expositions, University of Toronto Press: Toronto, 8 (1953).
15. [15] M. Mursaleen, K. J. Ansari and A. Khan: On $(p, q)-$analogue of Bernstein Operators. Appl. Math. Comput., 266 (2015), 874-882,(Erratum to: On $(p, q)-$analogue of Bernstein Operators, Appl. Math. Comput., 266 (2015), 874- 882.
16. [16] M. A. Ozarslan and O. Duman: Approximation theorems by Meyer-König and Zeller type operators. Chaos Solitons and Fractals.
17. [17] C. Radu: Statistical approximation by some linear operators of discrete type. MisKolc Math. Notes, 9(1) (2008), 61-68.
18. [18] P. N. Sadjang: On the fundamental theorem of $(p, q)-$calculus and some $(p, q)-$Taylor formulas. arXiv:1309.3934 [math.QA].
19. [19] H. Sharma and C. Gupta: On $(p, q)-$generalization of Szasz-Mirakyan Kantorovich operators. Bollettino dell’Unione Matematica Italiana, 8(3) (2016), 213-222.
20. [20] H. Sharma: Note on approximation properties of generalized Durrmeyer operators. Mathematical Sciences, 6(1) (2012), 1-6.
21. [21] T. Trif: Meyer-König and Zeller Operators based on the $q-$integers. Rev. Anal. Numer. Theor. Approx., 29(2) (2000), 221-229.
22. [22] W. Meyer-König and K. Zeller: Bernsteinsche Potenzreihen. studia math., 19 (1960), 89-94.