Abstract and Applied Analysis

Approximation of Durrmeyer Type Operators Depending on Certain Parameters

Neha Malik, Serkan Araci, and Man Singh Beniwal

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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.

Article information

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

Received: 29 January 2017
Revised: 30 March 2017
Accepted: 30 April 2017
First available in Project Euclid: 16 June 2017

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Malik, Neha; Araci, Serkan; Beniwal, Man Singh. Approximation of Durrmeyer Type Operators Depending on Certain Parameters. Abstr. Appl. Anal. 2017 (2017), Article ID 5316150, 9 pages. doi:10.1155/2017/5316150. https://projecteuclid.org/euclid.aaa/1497578544

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  • A. Lupaş, “A q-analogue of the bernstein operator,” Seminar on numerical and statistical calculus, Nr. 9, University of Cluj-Napoca, 1987.
  • A. Aral, V. Gupta, and R. P. Agarwal, Applications of q-Calculus in Operator Theory, Springer, 2013.
  • V. Gupta and R. P. Agarwal, Convergence Estimates in Approximation Theory, Springer, 2014.
  • M. Açikgöz, D. Erdal, and S. Araci, “A New approach to $q$-Bernoulli numbers and $q$-Bernoulli polynomials related to $q$-Bernstein polynomials,” Advances in Difference Equations, vol. 2010, Article ID 951764, 9 pages, 2011.
  • S. Araci, E. Ağyüz, and M. Acikgoz, “On a $q$-analog of some numbers and polynomials,” Journal of Inequalities and Applications, vol. 2015, article 19, 9 pages, 2015.
  • V. Kac and P. Cheung, Quantum Calculus, Universitext, Springer, New York, NY, USA, 2002.
  • M. Mursaleen, K. J. Ansari, and A. Khan, “On $(p,q)$-analogue of Bernstein operators,” Applied Mathematics and Computation, vol. 266, pp. 874–882, 2015.
  • M. Mursaleen, K. J. Ansari, and A. Khan, “Erratum to $`$on (p, q)-analogue of Bernstein Operators' [Appl. Math. Comput. 266 (2015) 874–882],” Applied Mathematics and Computation, vol. 278, pp. 70-71, 2016.
  • M. Mursaleen and F. Khan, “Approximation by Kantorovichtype $(p,q)$-Bernstein-Schurer Operators,” 2015, https://arxiv.org/abs/1506.02492.
  • V. Gupta and A. Aral, “Bernstein Durrmeyer operators based on two parameters,” Facta Universitatis. Series: Mathematics and Informatics, vol. 31, no. 1, pp. 79–95, 2016.
  • A. Aral and V. Gupta, “$(p,q)$-type beta functions of second kind,” Advances in Operator Theory, vol. 1, no. 1, pp. 134–146, 2016.
  • S. Araci, U. G. Duran, M. Acikgoz, and H. M. Srivastava, “Acertain $(p,q)$-derivative operator and associated divided differences,” Journal of Inequalities and Applications, vol. 301, 2016.
  • V. Gupta, “$(p,q)$-Baskakov-Kantorovich operators,” Applied Mathematics and Information Sciences, vol. 10, no. 4, pp. 1551–1556, 2016.
  • N. Malik and V. Gupta, “Approximation by (p,$\,$q)-Baskakov-BETa operators,” Applied Mathematics and Computation, vol. 293, pp. 49–56, 2017.
  • G. V. Milovanović, V. Gupta, and N. Malik, “$(p,q)$-Beta functions and applications in approximation,” Boletin de la Sociedad Matematica Mexicana. Serie III, pp. 1–19, 2016.
  • H. Sharma, “On Durrmeyer-type generalization of $(p,q)$-Bern-stein operators,” Arabian Journal of Mathematics, vol. 5, no. 4, pp. 239–248, 2016.
  • T. Acar, “$(p,q)$-generalization of Szász-Mirakyan operators,” Mathematical Methods in the Applied Sciences, vol. 39, no. 10, pp. 2685–2695, 2016.
  • Z. Finta, “Approximation properties of $(p,q)$-bernstein type operato,” Acta Universitatis Sapientiae, Mathematica, vol. 8, no. 2, pp. 222–232, 2016.
  • T. Acar, A. Aral, and S. A. Mohiuddine, “On Kantorovich modification of $(p,q)$-Baskakov operators,” Journal of Inequalities and Applications, vol. 2016, article 98, 2016.
  • T. Acar, A. Aral, and S. A. Mohiuddine, “Approximation bybivariate $(p,q)$-Bernstein-Kantorovich operators,” Iranian Journal of Science and Technology, Transactions A: Science, pp. 1–8, 2016.
  • A. Aral, G. Ulusoy, and E. Deníz, “A new construction of Szász-Mirakyan operators,” Numerical Algorithms, pp. 1–14, 2017.
  • V. Gupta, “Some approximation properties of q-Durrmeyer operators,” Applied Mathematics and Computation, vol. 197, no. 1, pp. 172–178, 2008.
  • P. N. Sadjang, “On the $(p,q)$-Gamma and the $(p,q)$-Beta functions,” 2015,.
  • P. N. Sadjang, “On the fundamental theorem of $(p,q)$-calcu-lus and some $(p,q)$-Taylor formulas,”.3934.
  • V. Sahai and S. Yadav, “Representations of two parameter quan-tum algebras and $p,q$-special functions,” Journal of Mathematical Analysis and Applications, vol. 335, no. 1, pp. 268–279, 2007.
  • Z. Finta and V. Gupta, “Approximation by $q$-Durrmeyer operators,” Journal of Applied Mathematics and Computing, vol. 29, no. 1-2, pp. 401–415, 2009.
  • D. D. Stancu, “Approximation of functions by a new class of linear polynomial operators,” Revue Roumaine de Mathématique Pures et Appliquées, vol. 13, pp. 1173–1194, 1968.
  • V. Gupta, “$(p,q)$-genuine Bernstein Durrmeyer operators,” Bollettino dell'Unione Matematica Italiana, vol. 9, no. 3, pp. 399–409, 2016.
  • R. A. DeVore and G. G. Lorentz, Constructive Approximation, Springer, Berlin, Germany, 1993. \endinput