Approximation properties of bivariate generalized Baskakov-Kantorovich operators

Document Type : Research Paper

Authors

1 School of Mathematics, Thapar Institute of Engineering and Technology, Patiala-147004, India

2 Department of Mathematics, Indian Institute of Technology Roorkee-247667, India

Abstract

The purpose of this paper is to study the bivariate extension of the generalized Baskakov-Kantorovich operators and obtain results on the degree of approximation, Voronovskaja type theorems and their first order derivatives in polynomial weighted spaces. Furthermore, we illustrate the convergence of the bivariate operators to a certain function through graphics using Matlab algorithm. We also discuss the comparison of the convergence of the bivariate generalized Baskakov Kantorovich operators and the bivariate Sz'{a}sz-Kantorovich operators to the function through illustrations using Matlab.

Keywords

[1] A.M. Acu, T. Acar, C.V. Muraru and V.A. Radu, Some approximation properties by a class of bivariate operators, Math. Meth. Appl. Sci. 42 (2019), 5551–5565.
[2] O. Agratini, Bivariate positive operators in polynomial weighted spaces, Abstr. Appl. Anal. 2013 (2013), 8.
[3] P.N. Agrawal and M. Goyal, Generalized Baskakov Kantorovich operators, Filomat 31(19) (2017) 6131–6151.
[4] P.N. Agrawal, V. Gupta, and A. Sathish Kumar, Generalized Baskakov-Durrmeyer type operators, Rend. Circ. Mat. Palermo 63 (2014), no. 2, 193–209.
[5] F. Altomare and M. Campiti, Korovkin-Type Approximation Theory and its Applications, de Gruyter Studies in Mathematics, 17. Walter de Grutyer and Co., Berlin 1994.
[6] F. Altomare and M. Cappelletti Montano and V. Leonessa, On a generalization of Szasz-Mirakjan-Kantorovich operators, Results Math. 63 (2013), no. 3-4, 837–863.
[7] G.A. Anastassiou and S. Gal, Approximation Theory. Moduli of Continuity and Global Smoothness Preservation, Birkhauser, Boston, 2000.
[8] D. Barbosu, Some generalized bivariate Bernstein operators, Math. Notes (Miskolc) 1 (2000), no. 1, 3–10.
[9] P.L. Butzer and H. Berens, Semi-groups of Operators and Approximation, Springer, New York, 1967.
[10] O. Dogru and V. Gupta, Korovkin-type approximation properties of bivariate q−Meyer-Konig and Zeller operators, Calcolo 43 (2006), no. 1, 51–63.
[11] A. Eren´╝îcin, Durrmeyer type modification of generalized Baskakov operators, Appl. Math. Comput. 218 (2011), no. 8, 4384–4390.
[12] B. Firlej and L. Rempulska, Approximation of functions of several variables by some operators of the Szasz-Mirakjan type, Fasc. math. 27 (1997) 15–27.
[13] M. Gurdek, L. Rempulska and M. Skorupka, The Baskakov operators for functions of two variables, Collect. Math. 50 (1999), no. 3, 289–302.
[14] A. Kajla and M. Goyal, Modified Bernstein-Kantorovich operators for functions of one and two variables, Rend. Circ. Mat. di Palermo II 67 (2018), no. 2, 379–395.
[15] V. Mihesan, Uniform approximation with positive linear operators generated by generalized Baskakov method, Automat. Comput. Appl. Math. 7 (1998), no. 1, 34–37.
[16] M. Nasiruzzaman, Approximation properties by Szasz-Mirakjan operators to bivariate functions via Dunkl analogue, Iran. J. Sci. Technol. Trans. A, Sci. 45 (2021), 259–269.
[17] F. Ozger, Weighted statistical approximation properties of univariate and bivariate λ-Kantorovich operators. Filomat 33 (2019), no. 11, 3473–3486.
[18] L. Rempulska and M. Skorupka, On convergence of first derivatives of certain Szasz-Mirakyan type operators, Rend. Mat. Appl. 19 (1999), no. 2, 269–279.
[19] M. Skorupka, Approximation of functions of two variables by some linear positive operators, Matematiche (Catania) 50 (1995), no. 2, 323–336.
[20] DD. Stancu, A new class of uniform approximating polynomial operators in two and several variables. G. Alexits, S. B. Stechkin (eds.): Proc. Conf. Consecutive Theory Functions, Budapest: Akad. Kiado, 1972, pp. 443–455.
[21] A. Wafi and S. Khatoon, Approximation by generalized Baskakov operators for functions of one and two variables in exponential and polynomial weight spaces, Thai. J. Math. 2 (2004) 53–66.
[22] A. Wafi and S. Khatoon, Convergence and Voronovskaja-type theorems for derivatives of generalized Baskakov operators, Cent. Eur. J. Math. 6 (2008), no. 2, 325–334.
[23] R. Yadav, R. Meher and V.N. Mishra, Results on bivariate Szasz-Mirakjan type operators in polynomial weight spaces, Math. Eng. Sci. Aerosp. 11 (2020), no. 4, 939–957.
[24] R. Yadav, R. Meher and V.N. Mishra, Quantitative estimations of bivariate summation-integral-type operators, Math. Meth. Appl. Sci. 42 (2019), 7172–7191.
[25] G. You and P. Xaun, Weighted approximation by multidimensional Baskakov operators, J. Math. Res. Expos. 20 (2000), no. 1, 43–50.
Volume 14, Issue 10
October 2023
Pages 361-375
  • Receive Date: 13 January 2023
  • Accept Date: 06 March 2023