Abstract
The present paper deals with the Stancu-type generalization of (p, q)-Baskakov–Durrmeyer operators. We investigate local approximation, weighted approximation properties of new operators and present the rate of convergence by means of suitable modulus of continuity. At the end of the paper, we introduce a new modification of (p, q)-Baskakov–Durrmeyer–Stancu operators with King approach.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Acar, T.: \((p, q)\)-generalization of Szász–Mirakyan operators. Math. Methods Appl. Sci. 39(10), 2685–2695 (2016)
Acar, T., Agrawal, P.N., Kumar. S.: On a modification of (p,q)-Szász-Mirakyan operators. Complex Anal. Oper. Theory. (2016). doi:10.1007/s11785-016-0613-9
Acar, T., Aral, A., Mohiuddine, S.A.: On Kantorovich modifications of \((p, q)\)-Baskakov operators. J. Inequal. Appl. 98, 2016 (2016)
Acar, T., Aral, A., Mohiuddine, S.A.: Approximation By Bivariate (p,q)-Bernstein-Kantorovich operators, Iran. J. Sci. Technol. Trans. Sci. (2016). doi:10.1007/s40995-016-0045-4
Aral, A., Gupta, V.: \((p, q)\)-type beta functions of second kind. Adv. Oper. Theor. 1(1), 134–146 (2016)
Aral, A., Gupta, V., Agarwal, R.P.: Applications of q-Calculus in Operator Theory. Springer, New York (2013)
Burban, I.: Two-parameter deformation of the oscillator albegra and \((p, q) \) analog of two dimensional conformal field theory. Nonlinear Math. Phys. 2(3–4), 384–391 (1995)
Burban, I.M., Klimyk, A.U.: \(P, Q\) differentiation, \(P, Q\) integration and \(P, Q\) hypergeometric functions related to quantum groups. Integr. Transforms Spec. Fucnt. 2(1), 15–36 (1994)
Devore, R.A., Lorentz, G.G.: Constructive Approximation. Springer, Berlin (1993)
Ibikli, E., Gadjieva, E.A.: The order of approximation of some unbounded function by the sequences of positive linear operators. Turk. J. Math. 19(3), 331–337 (1995)
Ilarslan, H.G.I., Acar, T.: Approximation by bivariate (p,q)-Baskakov–Kantorovich operators, Georgian Mathemath. J. (2016). doi:10.1515/gmj-2016-0057
King, J.P.: Positive linear operators which preserves \(x^{2}\). Acta Math. Hung. 99, 203–208 (2003)
Lopez-Moreno, A.J.: Weighted simultaneous approximation with Baskakov type operators. Acta Math. Hung. 104(1–2), 143–151 (2004)
Lupas, A.: A \(q\)-analogue of the Bernstein operator. Semin. Numer. Stat. Calc. Univ. Cluj Napoca 9, 85–92 (1987)
Mursaleen, M., Ansari, K.J., Khan, A.: On \((p, q)\)-analogue of Bernstein operators. Appl. Math. Comput. 266, 874–882 (2015)
Mursaleen, M., Ansari, K.J., Khan, A.: Some approximation results by \((p,q)\)-analogue of Bernstein–Stancu operators. Appl. Math. Comput. 264, 392–402 (2015) [Corrigendum: Appl. Math. Comput, 269, 744–746 (2015)]
Mursaleen, M., Nasiruzzaman, Md, Khan, A., Ansari, K.J.: Some approximation results on Bleimann–Butzer–Hahn operators defined by \((p, q)\)-integers. Filomat 30, 639–648 (2016)
Mursaleen, M., Nasiuzzaman, Md, Nurgali, A.: Some approximation results on Bernstein–Schurer operators defined by \((p, q)\)-integers. J. Inequal. Appl. 2015, 249 (2015)
Sahai, V., Yadav, S.: Representations of two parameter quantum algebras and \(p, q\)-special functions. J. Math. Anal. Appl. 335, 268–279 (2007)
Acknowledgements
The authors gratefully acknowledge the financial support from King Abdulaziz University, Jeddah, Saudi Arabia.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Palle Jorgensen.
Rights and permissions
About this article
Cite this article
Acar, T., Mohiuddine, S.A. & Mursaleen, M. Approximation by (p, q)-Baskakov–Durrmeyer–Stancu Operators. Complex Anal. Oper. Theory 12, 1453–1468 (2018). https://doi.org/10.1007/s11785-016-0633-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11785-016-0633-5