Generalized blending type Bernstein operators based on the shape parameter λ

In the present paper, we construct a new class of operators based on new type Bézier bases with a shape parameter λ and positive parameter s. Our operators include some well-known operators, such as classical Bernstein, α-Bernstein, generalized blending type α-Bernstein and λ-Bernstein operators as special case. In this paper, we prove some approximation theorems for these operators. Approximation properties of our operators are illustrated on graphs for variables s, α, λ, and n. It should be mentioned that our operators for λ=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lambda =1$\end{document} have better approximation than Bernstein and α-Bernstein operators.


Introduction
In 1912, Bernstein constructed Bernstein polynomials to prove Weierstrass Approximation Theorem [28], which says, for any continuous function f (x) on the closed interval where b n,k (x) = n k and n k = ⎧ ⎨ ⎩ n! (n-k)!k! , if 0 ≤ k ≤ n, 0, otherwise. (3) © The Author(s) 2022. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. [15,17,24], and references to many related works are also cited there. Later, Chen et al. (see [13]) extended Bernstein operators to α-Bernstein operators with a parameter α ∈ [0, 1], which are defined as

Many extensions of Bernstein operators have been given in
where p (α) 1,0 (x) = 1x, p (α) 1,1 (x) = x, and The α-Bernstein operators and their modifications have been intensively studied by many researchers in recent papers (see [1, 3-5, 11, 21, 23]). More recently, Aktuğlu et al. (see [3]) introduced and studied generalized blending type α-Bernstein operators by and which depend on two parameters α and s, where s is a positive integer, α One can see that when s = 1 and s = 2, then operators given by (5) and (6) reduce to ordinary Bernstein operators given by (1) and α-Bernstein operators given by (4), respectively. Aktuğlu and Yashar (see [4]) initiated and investigated some properties of generalized parametric blending type Bernstein operators that depend on four parameters s 1 , s 2 , a 1 , and a 2 .
The main purpose of the present paper is to construct a generalization of blending type Bernstein operators based on new type Bézier bases with a shape parameter λ and positive parameter s. A Korovkin-type approximation theorem will be proven. Moreover, approximation properties will also be discussed. For fixed s, α, λ, n, and specific function, detailed graphs will be given.
In order to make the calculations easier, we will use the following representation of L (α,s) n,λ (f ; x). For any 0 ≤ α ≤ 1, -1 ≤ λ ≤ 1 and a positive integer s, Here, B n,λ (f ; x) is given by equation (8), and B α,s n,λ (f ; x) is defined by where Proof Considering the definition of B n,λ (f ; x), we have Hence, which completes the proof of the first part.
Using the same techniques, B s,( ) n,λ (f ; x) can be proved in a similar way.

Lemma 1
(i) (Linearity) The (α, λ, s)-Bernstein operators are satisfying the following equality: where a 1 , a 2 are real numbers, and f (x) and g(x) are defined on the closed interval Remark 1 The operators L (α,s) n,λ (f ; x) have the following special cases: a) If α = 1 or s = 1, then L (α,s) n,λ (f ; x) reduces to the operators given in [10].
and a positive integer s, we have the following inequality: where · represents the uniform norm on C[0, 1]. Proof

Approximation properties of (α, λ, s)-Bernstein operators
This section is devoted to the approximation properties of the operators L (α,s) n,λ (f ; x). In this section, we will prove a Korovkin-type approximation theorem and approximation theorems by means of modulus of continuity and the Lipschitz function. n(n-1) ] if n < s, It is easily to see that for each case, L (α,s) n,λ (t; x) and L (α,s) n,λ (t 2 ; x) converge uniformly to e 1 (x) = x and e 2 (x) = x 2 , respectively. This completes the proof.
We use modulus of continuity to give quantitative error estimates for (α, λ, s)-Bernstein operators. We denote the usual modulus of continuity for f ∈ C[0, 1] as where ω is the usual modulus of the continuity.
Proof Since L (α,s) n,λ (1; x) = 1 andb α,s n,k (λ; x) ≥ 0 on [0, 1], we can write If we use the following properties of the modulus of continuity and where γ is a positive constant, we obtain Consequently, we can write If we apply the Cauchy-Schwarz inequality, we get n k=0b α,s So, we have Choosing δ = [ n,s,2 (x; α)] 1 2 , we complete the proof. Proof We have the following equality by applying the mean value theorem of differential calculus: where c = c n,k (x) ∈ (x, k n ). If we multiply both sides of the above equality byb α,s n,k (λ; x) and sum from 0 to n, we get Equivalently, Therefore, we can write Here, we observe that where δ is any positive number, which does not depend on k, and ω is the usual modulus. Consequently, we get Using the Cauchy-Schwarz inequality + ω f ; δ 1 + n,s,2 (x; α) δ n,s,2 (x; α) and choosing δ = n,s,2 (x; α), we complete the proof.
The Petree K -functional is given by It is given in [14] that there exists C > 0 such that where the second order modulus of continuity of smoothness for f ∈ C[0, 1] is defined as Now, we can prove the following theorem: , and s is arbitrary positive integer, then where C is a positive constant.

Conclusion remarks
In the present research paper, we introduce the operators L (α,s) n,λ (f ; x). Our operators L (α,s) n,λ (f ; x) are based on new type Bézier bases with a shape parameter λ and positive parameter s. Moreover, operators L (α,s) n,λ (f ; x) include classical Bernstein, α-Bernstein, generalized blending type α-Bernstein and λ-Bernstein operators as a special case. It should be mentioned that for λ = 0 and s = 2, our operators reduce to the operators defined by Chen et al. [13]. In this paper, some approximation properties of L (α,s) n,λ (f ; x) are proved and also are illustrated by graphical representations (see Fig. 1, Fig. 2, Fig. 3, Fig. 4). Our operators L (α,s) n,λ (f ; x) have better approximation for λ = 1 (see Fig. 2). Therefore, our operators have better approximation in comparison with the operators suggested and studied by Chen et al. (see Fig. 2). Finally, it should be mentioned that since our operators have better approximation for λ = 1, it gives better approximation than the other operators that can be obtained from our operators for λ = 0. For example, taking λ = 0 and α = 1, our operators reduce to Bernstein operators, and our operators for α = 1 and λ = 1 give better approximation than Bernstein operators.