q-Durrmeyer operators based on Pólya distribution

Volume 9, Issue 4, pp 1497--1504 http://dx.doi.org/10.22436/jnsa.009.04.08

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Authors

Vijay Gupta - Department of Mathematics, Netaji Subhas Institute of Technology Sector 3 Dwarka, New Delhi-110078, India. Themistocles M. Rassias - Department of Mathematics, National Technical University of Athens, Zografou Campus, 157 80, Athens, Greece. Honey Sharma - Department of Applied Sciences, Gulzar Group of Institutes, Khanna, Ludhiana, Punjab, India.

Abstract

We introduce a q analogue of Durrmeyer type modification of Bernstein operators based on Pólya distributions. We study the ordinary approximation properties of operators using modulus of continuity and Peetre K-functional of second order. Further, we establish the weighted approximation properties for these operators.

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Keywords

• Pólya distribution
• q-integers
• q-Bernstein operators
• modulus of continuity
• Peetre K-functional
• weighted modulus of continuity.

MSC

•  41A25
•  41A35

References

• [1] A. Aral, V. Gupta, R. P. Agarwal , Applications of q Calculus in Operator Theory, Springer, New York (2013)


• [2] I. Büyükyazıcı, H. Sharma, Approximation properties of two-dimensional q-Bernstein-Chlodowsky-Durrmeyer operators, Numer. Funct. Anal. Optim., 33 (2012), 1351-1371.


• [3] R. A. De Vore, G. G. Lorentz, Constructive Approximation, Springer-Verlag, Berlin (1993)


• [4] Z. Ditzian, V. Totik, Moduli of Smoothness, Springer-Verlag, New York (1987)


• [5] Z. Finta, V. Gupta, Approximation by q-Durrmeyer operators, J. Appl. Math. Comput., 29 (2009), 401-415.


• [6] V. Gupta, Some approximation properties of q-Durrmeyer operators, Appl. Math. Comput., 197 (2008), 172-178.


• [7] V. Gupta, Z. Finta, On certain q-Durrmeyer type operators, Appl. Math. Comput., 209 (2009), 415-420.


• [8] V. Gupta, W. Heping, The rate of convergence of q-Durrmeyer operators for 0 < q < 1, Math. Methods Appl. Sci., 31 (2008), 1946-1955.


• [9] V. Gupta, T. M. Rassias, Lupaş-Durrmeyer operators based on Pólya distribution, Banach J. Math. Anal., 8 (2014), 146-155.


• [10] V. Gupta, H. Sharma, Recurrence formula and better approximation for q-Durrmeyer Operators, Lobachevskii J. Math., 32 (2011), 140-145.


• [11] V. Gupta, H. Sharma, T. Kim, S. Lee, Properties of q-analogue of Beta operator, Adv. Difference Equ., 2012 (2012), 16 pages.


• [12] A. Lupaş , A q-analogue of the Bernstein operator, Seminar on Numerical and Statistical Calculus, Cluj-Napoca, 9 (1987), 85-92.


• [13] G. Nowak, Approximation properties for generalized q-Bernstein polynomials, J. Math. Anal. Appl., 350 (2009), 50-55.


• [14] G. Nowak, V. Gupta, The rate of pointwise approximation of possitive linear operators based on q-integer, Ukrainian Math. J., 63 (2011), 403-415.


• [15] G. M. Phillips, Bernstein polynomials based on the q-integers, Ann. Numer. Math., 4 (1997), 511-518.


• [16] D. D. Stancu, Approximation of functions by a new class of linear polynomial operators, Rev. Roumaine Math. Pures Appl., 13 (1968), 1173-1194.