Kantorovich variant of a new kind of q‐Bernstein–Schurer operators. (27th April 2017)

• Record Type: Journal Article
• Authors: Ruchi, Ruchi; Ispir, Nurhayat; Agrawal, P. N.
• Is Part Of: Mathematical methods in the applied sciences. Volume 40:Number 16(2017)
• Page Start: 5687
• Page End: 5706
• Publication Date: 2017-04-27
• Subjects: q‐Bernstein–Schurer–Kantorovich operators -- partial moduli of continuity -- complete modulus of continuity -- Bögel continuous -- Bögel differentiable -- GBS operators -- modulus of smoothness -- subclass 26A15 -- 41A25 -- 41A36; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.4418
• Languages: English
• ISSNs: 0170-4214

Abstract

Abstract : Ren and Zeng (2013) introduced a new kind of q ‐Bernstein–Schurer operators and studied some approximation properties. Acu et al . (2016) defined the Durrmeyer modification of these operators and studied the rate of convergence and statistical approximation. The purpose of this paper is to introduce a Kantorovich modification of these operators by using q ‐Riemann integral and investigate the rate of convergence by means of the Lipschitz class and the Peetre's K ‐functional. Next, we introduce the bivariate case of q ‐Bernstein–Schurer–Kantorovich operators and study the degree of approximation with the aid of the partial modulus continuity, Lipschitz space, and the Peetre's K ‐functional. Finally, we define the generalized Boolean sum operators of the q ‐Bernstein–Schurer–Kantorovich type and investigate the approximation of the Bögel continuous and Bögel differentiable functions by using the mixed modulus of smoothness. Furthermore, we illustrate the convergence of the operators considered in the paper for the univariate case and the associated generalized Boolean sum operators to certain functions by means of graphics using Maple algorithms. Copyright © 2017 John Wiley & Sons, Ltd.