Approximation degree of bivariate Kantorovich Stancu operators

Volume 14, Issue 6, pp 423--439 http://dx.doi.org/10.22436/jnsa.014.06.05

Publication Date: May 20, 2021 Submission Date: November 24, 2020 Revision Date: December 22, 2020 Accteptance Date: April 07, 2021

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Authors

P. N. Agrawal - Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, India. Neha Bhardwaj - Department of Applied Mathematics, Amity Institute of Applied Sciences, Amity University Uttar Pradesh, Noida, India. Jitendra Kumar Singh - Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, India.

Abstract

Abel et al. [U. Abel, M. Ivan, R. Paltanea, Appl. Math. Comput., \(\bf 259\) (2015), 116--123] introduced a Durrmeyer type integral variant of the Bernstein type operators based on two parameters defined by Stancu [D. D. Stancu, Calcolo, \(\bf 35\) (1998), 53--62]. Kajla [A. Kajla, Appl. Math. Comput., \(\bf 316\) (2018), 400--408] considered a Kantorovich modification of the Stancu operators wherein he studied some basic convergence theorems and also the rate of \(A\)-statistical convergence. In the present paper, we define a bivariate case of the operators proposed in [A. Kajla, Appl. Math. Comput., \(\bf 316\) (2018), 400--408] to study the degree of approximation for functions of two variables. We obtain the rate of convergence of these bivariate operators by means of the complete modulus of continuity, the partial moduli of continuity and the Peetre's \(K\)-functional. Voronovskaya and Gruss Voronovskaya type theorems are also established. We introduce the associated GBS (Generalized Boolean Sum) operators of the bivariate operators and discuss the approximation degree of these operators with the aid of the mixed modulus of smoothness for Bogel continuous and Bogel differentiable functions.

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Keywords

• Modulus of continuity
• Peetre's \(K\)-functional
• GBS operator
• B-continuous function
• mixed modulus of smoothness

MSC

•  41A10
•  41A25
•  41A30
•  41A63
•  26A15

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