Quantitative estimates for some modified Bernstein-Stancu operators

Abstract

In the papers [11,12] starting with the Bernstein operators, some Stancu type operators are constructed C n W Y !˘n .C n f /.x/ D n

Some approximation properties by a class of bivariate operators

“…The Bernstein operators of degree n are defined by x k (1 − x) n−k , k = 0, n, and b n,k (x) = 0, if k < 0 or k > n. Over time, many mathematicians start to generalize and modify classical Bernstein operators. For example, the reader can consult papers in previous studies [1][2][3][4][5][6] and the book by Gupta et al, 7 in which Gupta et al proposed a collection of several results concerning the Bernstein operator. The authors provide approximation properties, Voronovskaya-type theorems, or direct and inverse results for this modified or generalized operators.…”

Simultaneous Approximation for an Exponential Operator Connected with x4/3

“…Although there have already been several extensions and modifications of these type of operators in the existing research, approximation theory still finds this to be an interesting field of study. We mention some of the previous works here, see for instance [1,2,3,7,8,10,11,12,13]. In [5], the authors examined an operator of the exponential-type associated with 𝑝(𝑥) = 𝑥 4/3 and proved its direct estimates, quantitative variants of the Voronovskaja theorem and several convergence estimates.…”

Approximation by generalized Bernstein–Stancu operators

“…Also, in 2007, Gonska et al [18] proved convergence results for over-iterates of certain (generalized) Bernstein-Stancu operators. Using King's technique in modifying the positive linear operators, in [26] the authors obtained a new class starting with the Bernstein-Stancu type operators investigated in [13,31]. We remark that recently the study of Bernstein-Stancu type operators was extended in new directions as we can see, for example, in the works of [8,10,11,16,20,25,29,32].…”