Approximation of Durrmeyer type operators depending on certain parameters. (English) Zbl 1470.41012

Malik, Neha; Araci, Serkan; Beniwal, Man Singh

Abstr. Appl. Anal. 2017, Article ID 5316150, 9 p. (2017).

Summary: Motivated by a number of recent investigations, we consider a new analogue of Bernstein-Durrmeyer operators based on certain variants. We derive some approximation properties of these operators. We also compute local approximation and Voronovskaja type asymptotic formula. We illustrate the convergence of aforementioned operators by making use of the software MATLAB which we stated in the paper.

MSC:

41A35	Approximation by operators (in particular, by integral operators)
41A25	Rate of convergence, degree of approximation

Software:

Matlab

References:

[1]	A. Lupaş, A q-analogue of the bernstein operator, Seminar on numerical and statistical calculus, Nr. 9, (1987), University of Cluj-Napoca · Zbl 0684.41014
[2]	Aral, A.; Gupta, V.; Agarwal, R. P., Applications of q-Calculus in Operator Theory, (2013), Springer · Zbl 1273.41001
[3]	Gupta, V.; Agarwal, R. P., Convergence Estimates in Approximation Theory, (2014), Springer · Zbl 1295.41002
[4]	Açikgöz, M.; Erdal, D.; Araci, S., A New approach to \(q\)-Bernoulli numbers and \(q\)-Bernoulli polynomials related to \(q\)-Bernstein polynomials, Advances in Difference Equations, 2010, (2011) · Zbl 1229.11034
[5]	Araci, S.; Ağyüz, E.; Acikgoz, M., On a \(q\)-analog of some numbers and polynomials, Journal of Inequalities and Applications, 2015, article 19, (2015) · Zbl 1308.05020
[6]	Kac, V.; Cheung, P., Quantum Calculus. Quantum Calculus, Universitext, (2002), New York, NY, USA: Springer, New York, NY, USA · Zbl 0986.05001
[7]	Mursaleen, M.; Ansari, K. J.; Khan, A., On \(\left(p, q\right)\)-analogue of Bernstein operators, Applied Mathematics and Computation, 266, 874-882, (2015) · Zbl 1410.41004
[8]	Mursaleen, M.; Ansari, K. J.; Khan, A., Erratum to ‘on (p, q)-analogue of Bernstein Operators’ [Appl. Math. Comput. 266 (2015) 874–882], Applied Mathematics and Computation, 278, 70-71, (2016) · Zbl 1409.41005
[9]	Mursaleen, M.; Khan, F., Approximation by Kantorovich type \(\left(p, q\right)\)-Bernstein-Schurer Operators · Zbl 1355.41018
[10]	Gupta, V.; Aral, A., Bernstein Durrmeyer operators based on two parameters, Facta Universitatis. Series: Mathematics and Informatics, 31, 1, 79-95, (2016) · Zbl 1474.41036
[11]	Aral, A.; Gupta, V., \(\left(p, q\right)\)-type beta functions of second kind, Advances in Operator Theory, 1, 1, 134-146, (2016) · Zbl 1359.41004
[12]	Araci, S.; Duran, U. G.; Acikgoz, M.; Srivastava, H. M., A certain \(\left(p, q\right)\)-derivative operator and associated divided differences, Journal of Inequalities and Applications, 301, (2016) · Zbl 1393.05059
[13]	Gupta, V., \(\left(p, q\right)\)-Baskakov-Kantorovich operators, Applied Mathematics and Information Sciences, 10, 4, 1551-1556, (2016)
[14]	Malik, N.; Gupta, V., Approximation by (p, q)-Baskakov-BETa operators, Applied Mathematics and Computation, 293, 49-56, (2017) · Zbl 1411.41016
[15]	Milovanović, G. V.; Gupta, V.; Malik, N., \(\left(p, q\right)\)-Beta functions and applications in approximation, Boletin de la Sociedad Matematica Mexicana. Serie III, 1-19, (2016) · Zbl 1387.33003
[16]	Sharma, H., On Durrmeyer-type generalization of \(\left(p, q\right)\)-Bernstein operators, Arabian Journal of Mathematics, 5, 4, 239-248, (2016) · Zbl 1357.41016
[17]	Acar, T., \((p, q)\)-generalization of Szász-Mirakyan operators, Mathematical Methods in the Applied Sciences, 39, 10, 2685-2695, (2016) · Zbl 1342.41019
[18]	Finta, Z., Approximation properties of \(\left(p, q\right)\)-bernstein type operato, Acta Universitatis Sapientiae, Mathematica, 8, 2, 222-232, (2016) · Zbl 1356.41011
[19]	Acar, T.; Aral, A.; Mohiuddine, S. A., On Kantorovich modification of \(\left(p, q\right)\)-Baskakov operators, Journal of Inequalities and Applications, 2016, article 98, (2016) · Zbl 1333.41007
[20]	Acar, T.; Aral, A.; Mohiuddine, S. A., Approximation by bivariate \((p, q)\)-Bernstein-Kantorovich operators, Iranian Journal of Science and Technology, Transactions A: Science, 1-8, (2016) · Zbl 1397.41006
[21]	Aral, A.; Ulusoy, G.; Deníz, E., A new construction of Szász-Mirakyan operators, Numerical Algorithms, 1-14, (2017) · Zbl 1408.41016
[22]	Gupta, V., Some approximation properties of q-Durrmeyer operators, Applied Mathematics and Computation, 197, 1, 172-178, (2008) · Zbl 1142.41008
[23]	Sadjang, P. N., On the \(\left(p, q\right)\)-Gamma and the \(\left(p, q\right)\)-Beta functions
[24]	Sadjang, P. N., On the fundamental theorem of \(\left(p, q\right)\)-calculus and some \(\left(p, q\right)\)-Taylor formulas · Zbl 1387.30080
[25]	Sahai, V.; Yadav, S., Representations of two parameter quantum algebras and \(p, q\)-special functions, Journal of Mathematical Analysis and Applications, 335, 1, 268-279, (2007) · Zbl 1141.17015
[26]	Finta, Z.; Gupta, V., Approximation by \(q\)-Durrmeyer operators, Journal of Applied Mathematics and Computing, 29, 1-2, 401-415, (2009) · Zbl 1198.41008
[27]	Stancu, D. D., Approximation of functions by a new class of linear polynomial operators, Revue Roumaine de Mathématique Pures et Appliquées, 13, 1173-1194, (1968) · Zbl 0167.05001
[28]	Gupta, V., \(\left(p, q\right)\)-genuine Bernstein Durrmeyer operators, Bollettino dell’Unione Matematica Italiana, 9, 3, 399-409, (2016) · Zbl 1350.33002
[29]	DeVore, R. A.; Lorentz, G. G., Constructive Approximation, (1993), Berlin, Germany: Springer, Berlin, Germany · Zbl 0797.41016