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Kantorovich variant of a new kind of q ‐Bernstein–Schurer operators
Author(s) -
Ruchi Ruchi,
Ispir Nurhayat,
Agrawal P. N.
Publication year - 2017
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.4418
Subject(s) - mathematics , modulus of continuity , bivariate analysis , rate of convergence , bernstein polynomial , univariate , lipschitz continuity , operator theory , type (biology) , pure mathematics , smoothness , discrete mathematics , mathematical analysis , multivariate statistics , ecology , channel (broadcasting) , statistics , electrical engineering , biology , engineering
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.