Shape-preserving properties of a new family of generalized Bernstein operators

Abstract

In this paper, we introduce a new family of generalized Bernstein operators based on q integers, called -Bernstein operators, denoted by . We investigate a Kovovkin-type approximation theorem, and obtain the rate of convergence of to any continuous functions f. The main results are the identification of several shape-preserving properties of these operators, including their monotonicity- and convexity-preserving properties with respect to . We also obtain the monotonicity with n and q of .

Bivariate α,q-Bernstein–Kantorovich Operators and GBS Operators of Bivariate α,q-Bernstein–Kantorovich Type

“…In order to prove the main conclusion of this paper, the following lemmas are given: Lemma 1. (See [3]) The following equalities hold:…”

“…n,i (x) for i = 0, 1, ..., n are defined in Equation (2). Very recently, Cai and Xu [3] proposed the α, q-Bernstein operators as…”

On the eigenstructure of the $(α,q)$-Bernstein operator

“…Eigenfunctions of the q-Bernstein operators and their asymptotic behavior are investigated by S. Ostrovska and M.Turan in 2013 [16]. In this study, eigenvalues and eigenfunctions of the (α, q)-Bernstein operators introduced by Qing-Bo Cai and Xiao-Wei Xu in [18] are found and their asymptotic behavior is investigated. When q = 1, one obtains the eigenvalues and eigenfunctions of the α-Bernstein operator introduced by Chen et.…”

Approximation by α-Bernstein-Schurer operator

“…Also, they gave an upper bound for the approximation error by means of the modulus of continuity and proved that the α-Bernstein operators satisfy some shape preserving results. Then, many researchers have studied intensively α−Bernstein operators and their generalizations for the last two years (see, e.g., [1,2,4,7,10,11,17,20]).…”