Statistical approximation properties of λ-Bernstein operators based on q-integers

Abstract

In this paper, we introduce a new generalization of λ-Bernstein operators based on q-integers, we obtain the moments and central moments of these operators, establish a statistical approximation theorem and give an example to show the convergence of these operators to f(x). It can be seen that in some cases the absolute error bounds are smaller than the case of classical q-Bernstein operators to f(x).

Convergence of λ-Bernstein operators based on (p, q)-integers

“…Later, Özger [3] gave some statistical approximation results of (1). In 2019, Cai et al [4] proposed new λ-Bernstein operators based on q-integers and established a statistical approximation theorem. Some other papers also mention λ-Bernstein operators, see [5,6].…”

“…Obviously, when p → 1 -, B λ n,p,q (f ; x) reduces to q-analogue of λ-Bernstein operators in [4]; when p, q → 1 -, B λ n,p,q (f ; x) reduces to (1). Here we mention certain notations on (p, q)-calculus, details can be found in [23][24][25][26][27].…”

On new Bézier bases with Schurer polynomials and corresponding results in approximation theory

“…A new type -Bernstein operators have been introduced by Cai et al in [6] based on Bézier bases de…ned by Ye et al in [30]. We refer to [5,6,20,23,26] for recent studies about -Bernstein type operators and [13,14,28] for some Schuer type operators.…”

Note on a New Construction of Kantorovich Form q-Bernstein Operators Related to Shape Parameter λ

“…In 2019, Cai et al [20] proposed and studied some statistical approximation properties of a new generalization of (λ, q)-Bernstein polynomials via Bézier bases with shape parameter λ ∈ [−1, 1] as follows:…”