WEIGHTED APPROXIMATION PROPERTIES OF STANCU TYPE MODIFICATION OFq-SZÁSZ-DURRMEYER OPERATORS

Year 2016, Volume: 65 Issue: 1, 87 - 104, 01.02.2016

Gürhan Içöz N. Mohapatra R.

https://doi.org/10.1501/Commua1_0000000746

Cited By: 3

https://izlik.org/JA88NH28CC

Abstract

In this paper, we are dealing with q-Sz·sz-Mirakyan-DurrmeyerStancu operators. Firstly, we establish moments of these operators and estimate convergence results. We discuss a Voronovska ja type result for the
operators. We shall give the weighted approximation properties of these operators. Furthermore, we study the weighted statistical convergence for the
operators.

Keywords

q-Durrmeyer operators , q-Jackson integral , q-Beta functions

References

• Agratini O., Do¼gru O., Weighted statistical approximation by q-Szász type operators that preserve some test functions, Taiwanese J. Math., 2010, 14, 4, 1283-1296
• Aral A., Gupta V., The q-derivative and applications to q-Szász-Mirakyan operators, Calcolo , 43, 151-170
• Aral A., A generalization of Szász-Mirakyan operators based on q-integers, Math. Comput. Model. 2008, 47, 1052-1062
• Derriennic M. M., Modi…ed Bernstein polynomials and Jacobi polynomials in q-calculus, Rendiconti Del Circolo Matematico Di Palermo, Serie II, Suppl. 2005, 76, 269-290
• De Sole A., Kac V., On integral representations of q-gamma and q-beta functions, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 2005, 16, 11-29
• Do¼gru O., On statistical approximation properties of Stancu type bivariate generalization of q-Balazs-Szabados operators. In Proceedings of the International Conference on Numerical Analysis and Approximation Theory, University of Babes-Bolyai, Cluj-Napoca (5–8 July ) Finta Z., Gupta V., Approximation by q-Durrmeyer operators, J. Appl. Math. Comput., , 29, 401-415
• Gadjiev A. D., The convergence problem for a sequence of positive linear operators on un- bounded sets, and theorems analogous to that of P. P. Korovkin, Soviet Math. Dokl., 1974, (5), 1433–1436
• Gadjiev A.D., Orhan C., Some approximation theorems via statistical convergence, Rocky Mountain J. Math., 2002, 32, 1, 129-138
• Gupta V., Some approximation properties of q-Durrmeyer operators, Appl. Math. Comput., , 197, 172-178
• Gupta V., On q-Phillips operators, Georgian Math. J., submitted Gupta V., A note on modi…ed Szász operators, Bull. Inst. Math. Acad. Sinica, 1993, 21(3), 278
• Gupta V., Deo N., Zeng X., Simultaneous approximation for Szász-Mirakian-Stancu- Durrmeyer operators, Anal. Theory Appl., 2013, 29 (1), 86-96
• Gupta V., Aral A., Özhavzali M., Approximation by q-Szász-Mirakyan-Baskakov operators, Fasciculi Mathematici, 2012, 48, 35-48
• Gupta V., Heping W., The rate of convergence of q-Durrmeyer operators for 0 < q < 1, Math. Methods Appl. Sci., 2008, 31, 1946-1955
• Gupta V., Srivastava G. S., On the rate of convergence of Phillips operators for functions of bounded variation, Annal. Soc. Math. Polon. Comment. Math., 1996, 36, 123-130
• Gupta V., Karslı H., Some approximation properties by q-Szász-Mirakyan-Baskakov-Stancu operators, Lobachevskii Journal of Mathematics, 2012, 33(2), 175-182
• Jackson F. H., On q-de…nite integrals, Quart. J. Pure and Applied Math., 41, 1910, 193-203
• Kac V., Cheung P., Quantum Calculus, Universitext, Springer-Verlag, New York, 2002
• Mahmudov N. I., Approximation by the q-Szász-Mirakyan operators, Abstr. Appl. Anal., , Article ID 754217, doi:10.1155/2012/754217
• Mahmudov N. I., On q-parametric Szász-Mirakyan operators, Mediterranean J. Math., 2010, (3), 297-311
• Mahmudov N. I., Kağao¼glu H., On q-Szász-Durrmeyer operators, Cent. Eur. J. Math., 2010, , 399-409
• Mahmudov N. I., Gupta V., Kağao¼glu H., On certain q-Phillips operators, Rocky Mountain J. Math., 2012, 42, 4, 1291-1310
• May C. P., On Phillips operator, J. Approx. Theory, 1977, 20, 315-332
• Örkcü M., Do¼gru O., Weighted statistical approximation by Kantorovich type q-Szász- Mirakjan operators, Appl. Math. Comput., 2011, 217, 7913-7919
• Örkcü M., Do¼gru O., Statistical approximation of a kind of Kantorovich type q-Szász- Mirakjan operators, Nonlinear Anal-Theor., 2012, 75, 2874-2882
• Phillips G. M., Bernstein polynomials based on the q-integers, Ann. Numer. Math., 1997, 4, 518
• Phillips R. S., An inversion formula for Laplace transforms and semi-groups of linear oper- ators, Annals of Mathematics. Second Series, 1954, 59, 325–356
• Prasad G., Agrawal P. N., Kasana H. S., Approximation of functions on [0; 1) by a new sequence of modi…ed Szász operators, Math. Forum, 1983, 6(2), 1-11.
• Current address : Department of Mathematics, Gazi University, Ankara, Turkey; E-mail address : [e-mail]. Current address : Department of Mathematics, University of Central Florida, Orlando, FL, USA; E-mail address : [e-mail].