Kantrovich Type Generalization of Meyer-Konig and Zeller Operators via Generating Functions

Olgun, Ali; İnce, H. Gül; Tasdelen, Fatma

doi:10.2478/auom-2013-0053

Abstract

In the present paper, we study a Kantorovich type generalization of Meyer-König and Zeller type operators via generating functions. Using Korovkin type theorem we first give approximation properties of these operators defined on the space C [0;A] ; 0 < A < 1. Secondly, we compute the rate of convergence of these operators by means of the modulus of continuity and the elements of the modified Lipschitz class. Finally, we give an r-th order generalization of these operators in the sense of Kirov and Popova and we obtain approximation properties of them.

References

[1] A. Altin, O. Dogru, and F. Tasdelen, The generalization of Meyer-Konig and Zeller operators by generating functions, J. Math. Anal. Appl. 312, 181-194 (2005).
Search in Google Scholar
[2] O. Agratini, Korovkin type estimates for Meyer-Konig and Zeller operators, Math. Ineq. Appl. no. 1, 4, 119-126 (2001).
Search in Google Scholar
[3] E. W. Cheney and A. Sharma, A. Bersntein power series, Canad. J. Math. 16, 241-253 (1964).
Search in Google Scholar
[4] O. Dogru, Approximation order and asymtotic approximation for Generalized Meyer-Konig and Zeller operators, Math. Balkanica, N.S. no. 3-4, 12, 359-368 (1998).
Search in Google Scholar
[5] O. Dogru, and N. Ozalp, Approximation by Kantorovich type generalization of Meyer-Konig and Zeller operators, Glasnik Mat. no 36, 56 , 311-318 (2001).
Search in Google Scholar
[6] O. Doggru, O. Duman and C. Orhan, Statistical approximation by generalized Meyer-Konig and Zeller type operator, Studia Sci. Math. Hungar. no. 3, 40, 359-371 (2003).
Search in Google Scholar
[7] A. D. Gadjiev and O. (Dakar, On uniform approximation by Bleimann, Butzer and Hahn operators on all positive semi-axis, Trans. Acad. Sci. Azerb. Ser. Phys. Tech. Math. Sci. 19, 21-26 (1999).
Search in Google Scholar
[8] M. K. Khan, On the rate of convergence of Bersntein power series for functions of bounded variation, J. Approx. Theory, no. 1, 57, 90-103 (1989).
Search in Google Scholar
[9] G. Kirov and L. Popova, A generalization of the linear positive operators, Math. Balkanica 7, 149-162 (1993).
Search in Google Scholar
[10] P. P. Korovkin, Linear operators and approximation theory, Hindustan Publish Co., Delphi (1960).
Search in Google Scholar
[11] B. Lenze, Bersntein-Baskakov-Kantorovich operators and Lipschitz-type maximal functions, in: Approx. Th., Kecskemet, Hungary, Colloq. Math. Soc. Janos Bolyai 58, 469-496 (1990).
Search in Google Scholar
[12] A. Lupas and M. W. Miiller, Approximation properties of the Mn- operators, Aequationes Math. 5, 19-37 (1970).
Search in Google Scholar
[13] W. Meyer-Konig and K. Zeller, Bernsteinsche potenzreihen, Studia Math. 19, 89-94 (1960).
Search in Google Scholar
[14] H. N. Mhaskar and D. V. Pai, Fundamentals of approximation theory. CRC Press, Boca Raton, FL; Narosa Publishing House, New Delhi, 2000.
Search in Google Scholar
[15] M. A. Ozarslan, O. Duman and H. M. Srivastava, Statistical approximation results for Kantorovich type operators involving some special polynomials, Math. Comp. Mod. 48 (2008), 388-401.10.1016/j.mcm.2007.08.015
Search in Google Scholar
[16] A. I. Stepanets, Methods of approximation theory, VSP, Leiden, 2005. ISBN: 90-6764-427-7. 10.1515/9783110195286
Search in Google Scholar

Kantrovich Type Generalization of Meyer-Konig and Zeller Operators via Generating Functions

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