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Approximation of B-continuous and B-differentiable functions by GBS operators of q-Bernstein-Schurer-Stancu type

The concern of this paper is to obtain the rate of convergence in terms of the partial and complete modulus of continuity and the degree of approximation by means of Lipschitz-type class for the bivariate operators. | Turk J Math (2016) 40: 1298 – 1315 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ doi:10.3906/mat-1509-66 Research Article Approximation of B -continuous and B -differentiable functions by GBS operators of q -Bernstein–Schurer–Stancu type ˙ ˙ 2 , Purshottam Narain AGRAWAL1 Manjari SIDHARTH1,∗, Nurhayat ISP IR 1 Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, India 2 Department of Mathematics, Faculty of Sciences, Gazi University, Ankara, Turkey Received: 22.09.2015 • Accepted/Published Online: 13.02.2016 • Final Version: 02.12.2016 Abstract: Bˇ arbosu and Muraru (2015) introduced the bivariate generalization of the q -Bernstein–Schurer–Stancu operators and constructed a GBS operator of q -Bernstein–Schurer–Stancu type. The concern of this paper is to obtain the rate of convergence in terms of the partial and complete modulus of continuity and the degree of approximation by means of Lipschitz-type class for the bivariate operators. In the last section we estimate the degree of approximation by means of Lipschitz class function and the rate of convergence with the help of mixed modulus of smoothness for the GBS operator of q -Bernstein–Schurer–Stancu type. Furthermore, we show comparisons by some illustrative graphics in Maple for the convergence of the operators to some functions. Key words: q -Bernstein–Schurer–Stancu operators, partial moduli of continuity, B-continuous, B-differentiable, GBS operators, modulus of smoothness, degree of approximation 1. Introduction In 1987, q -based Bernstein operators were defined and studied by Lupas [21]. In 1997, another q-based Bernstein operator was proposed by Phillips [23]. Since then q-based operators have become an active research area. Muraru [22] introduced and investigated the q-Bernstein–Schurer operators. She obtained the Korovkin-type approximation theorem and the rate of convergence of the operators in terms of the first modulus of .