Approximation by Schurer Type λ-Bernstein–Bézier Basis Function Enhanced by Shifted Knots Properties

Abstract

In this article, a novel Schurer form of $\lambda$-Bernstein operators augmented by Bézier basis functions is presented by utilizing the features of shifted knots. The shifted knots form of Bernstein operators and the Schurer form of the Bézier basis function are used in this article, then, new operators, the Schurer type $\lambda$-Bernstein shifted knots operators are constructed in terms of the Bézier basis function. First, the test functions are calculated and the central moments for these operators are obtained. Then, Korovkin’s type approximation properties are studied by the use of a modulus of continuity of orders one and two. Finally, the convergence theorems for these new operators are obtained by using Peetre’s K-functional and Lipschitz continuous functions. In the end, some direct approximation theorems are also obtained.

1. Introduction and Preliminaries

One of the most well-known mathematicians in the world, S. N. Bernstein, provided the quickest and most elegant demonstration of one of the most well-known Weierstrass approximation theorems. Bernstein also devised the series of positive linear operators implied by ${\left\{B_s\right\}}_{s\ge 1}$. The famous Bernstein operators, defined in [1], were found to be a function that uniformly approximates every continuous function on $[0,1]$ that is $g\in C[0,1]$. These finding were made in Bernstein’s study. Thus, for any $y\in [0,1]$, the well-known Bernstein operators defined by

$$B_s(g;y)=\sum _{i=0}^{s}g\left({\displaystyle \frac{i}{s}}\right)b_{s,i}\left(y\right),$$

where the Bernstein polynomials with a maximum degree of s are expressed as $b_{s,i}\left(y\right)$ and $s\in \mathbb{N}$ (positive integers):

$$b_{s,i}\left(y\right)=\left({\displaystyle \genfrac{}{}{0pt}{}{s}{i}}\right)y^i{(1-y)}^{s-i}\left(i=0,1,\cdots ,s;y\in \left[0,1\right]\right)$$

and

$$b_{s,i}\left(y\right)=0(i>s\mathrm{or}i<0).$$

The Bernstein polynomials have an extremely simple recursive relation to check. For Bernstein polynomials $b_{s,i}\left(y\right)$, it is quite simple to prove the recursive relationship:

$$b_{s,i}\left(y\right)=(1-y)b_{s-1,i}\left(y\right)+y{b}_{s-1,i-1}\left(y\right).$$

By using a positive integer, the operators were defined and known as the Bernstein–Schurer operators. Suppose $S_{n,k}^{\mu }:C\left[0,1+\mu \right]\to C[0,1]$, and Schurer [2] presented modification of Bernstein operators in 1962 as follows:

$$S_{n,k}^{\mu }(g;y)=\sum _{k=0}^{n+\mu }g\left({\displaystyle \frac{k}{n}}\right)s_{n,\mu ,k}\left(y\right)(y\in [0,1]),$$

(1)

where $\mu$ is a fixed positive integer and $s_{n,\mu ,k}\left(y\right)$ is referred to as the basic Bernstein–Schurer polynomials so that

$$s_{n,\mu ,k}\left(y\right)=\left({\displaystyle \genfrac{}{}{0pt}{}{n+\mu }{k}}\right)y^k{(1-y)}^{n+\mu -k}(k=0,1,\cdots ,n+\mu ).$$

(2)

The Bernstein operators were introduced in 2010 by Cai et al. [3] by using $\lambda \in [-1,1]$. These operators are referred to as $\lambda$-Bernstein operators:

$$B_{s,\lambda }(g;y)=\sum _{i=0}^{s}g\left({\displaystyle \frac{i}{s}}\right){\tilde{b}}_{s,i}(\lambda ;y),$$

(3)

where the Bernstein polynomial $b_{s,i}\left(y\right)$ defined by Ye et al. [4] is expressed in terms of the new Bernstein basis function ${\tilde{b}}_{s,i}(\lambda ;y)$ as follows:

$$\begin{array}{ccc}& & {\tilde{b}}_{s,0}(\lambda ;y)=b_{s,0}\left(y\right)-{\displaystyle \frac{\lambda }{s+1}}{b}_{s+1,1}\left(y\right),\hfill \\ & & {\tilde{b}}_{s,i}(\lambda ;y)=b_{s,i}\left(y\right)+\lambda ({\displaystyle \frac{s-2i+1}{s^2-1}}{b}_{s+1,i}\left(y\right)\hfill \\ & & -{\displaystyle \frac{s-2i-1}{s^2-1}}{b}_{s+1,i+1}\left(y\right)),\mathrm{for}1\le i\le s-1\hfill \\ & & {\tilde{b}}_{s,s}(\lambda ;y)=b_{s,s}\left(y\right)-{\displaystyle \frac{\lambda }{s+1}}{b}_{s+1,s}\left(y\right).\hfill \end{array}$$

Cai [5] and Cai and Xu [6] have also introduced the Kantorovich form of the $\lambda$-Bernstein–Bézier operators and studied shape preserving properties of generalized Bernstein operators, respectively.

Using the shifted knots properties, the following are the recent Bernstein-type Stancu operators, which were first introduced by Gadjiev et al. in 2010 [7]:

$$S_{s,\mu ,\beta }(g;y)={\left({\displaystyle \frac{s+{\nu }_2}{m}}\right)}^m\sum _{i=0}^s\left({\displaystyle \genfrac{}{}{0pt}{}{s}{i}}\right){\left(y-{\displaystyle \frac{{\mu }_2}{s+{\nu }_2}}\right)}^i{\left({\displaystyle \frac{s+{\mu }_2}{m+{\nu }_2}}-y\right)}^{s-i}g\left({\displaystyle \frac{i+{\mu }_1}{s+{\nu }_1}}\right),$$

(4)

given that $y\in [{\displaystyle \frac{{\mu }_2}{m+{\nu }_2}},{\displaystyle \frac{s+{\mu }_2}{s+{\nu }_2}}]$, and given that $0\le {\mu }_2\le {\mu }_1\le {\nu }_1\le {\nu }_2$, ${\mu }_i,{\nu }_i,i=1,2$ are positive real numbers.

Through approximation procedures, researchers have recently developed Bernstein-type operators. Among them are Bernstein–Kantorovich–Stancu shifted knots operators [8], q-Bernstein operators associated with the $\lambda$ shape parameter [9], the Bernstein–Kantorovich operators in Stancu variation [10], the q-Bernstein–Stancu–Kantorovich operators [11], the Lupaş–Durrmeyer operators in Pólya distribution form [12], the new family of Bernstein–Kantorovich operators [13] and in a fractional sense [14], Bézier bases with Schurer polynomials [15], q-Bernstein shifted operators [16], $\alpha$-Bernstein–Schurer operators [17], and generalized Bernstein–Schurer operators [18]. Based on Bézier bases, there are Bernstein operators [19], approximations of Bernstein-type operators [20], and others such as [21,22,23].

Most recently, Mursaleen et al. [24] constructed the shifted knots of $\lambda$-Bernstein operators by applying Bézier bases functions. For all ${\displaystyle \frac{{\kappa }_1}{s+{\kappa }_2}}\le y\le {\displaystyle \frac{s+{\kappa }_1}{s+{\kappa }_2}}$ and the real number $0\le {\kappa }_1\le {\kappa }_2$, the $\lambda$-Bernstein shifted knots operators $B_{s,\lambda }^{{\kappa }_1,{\kappa }_2}$ were defined by:

$$B_{s,\lambda }^{{\kappa }_1,{\kappa }_2}(g;y)={\left({\displaystyle \frac{s+{\kappa }_2}{s}}\right)}^s\sum _{i=0}^s{\tilde{b}}_{s,i}^{{\kappa }_1,{\kappa }_2}(\lambda ;y)g\left({\displaystyle \frac{i}{s}}\right),$$

(5)

where

$$\begin{array}{ccc}& & {\tilde{b}}_{s,0}^{{\kappa }_1,{\kappa }_2}(\lambda ;y)=b_{s,0}^{{\kappa }_1,{\kappa }_2}\left(y\right)-{\displaystyle \frac{\lambda }{s+1}}{b}_{s+1,1}^{{\kappa }_1,{\kappa }_2}\left(y\right),\hfill \\ & & {\tilde{b}}_{s,i}^{{\kappa }_1,{\kappa }_2}(\lambda ;y)=b_{s,i}^{{\kappa }_1,{\kappa }_2}\left(y\right)+\lambda ({\displaystyle \frac{s-2i+1}{s^2-1}}{b}_{s+1,i}^{{\kappa }_1,{\kappa }_2}\left(y\right)\hfill \\ & & -{\displaystyle \frac{s-2i-1}{s^2-1}}{b}_{s+1,i+1}^{{\kappa }_1,{\kappa }_2}\left(y\right)),\mathrm{for}1\le i\le s-1\hfill \\ & & {\tilde{b}}_{s,s}^{{\kappa }_1,{\kappa }_2}(\lambda ;y)=b_{s,s}^{{\kappa }_1,{\kappa }_2}\left(y\right)-{\displaystyle \frac{\lambda }{s+1}}{b}_{s+1,s}^{{\kappa }_1,{\kappa }_2}\left(y\right);\hfill \end{array}$$

and

$$b_{s,i}^{{\kappa }_1,{\kappa }_2}\left(y\right)=\left({\displaystyle \genfrac{}{}{0pt}{}{s}{i}}\right){\left(y-{\displaystyle \frac{{\kappa }_1}{s+{\kappa }_2}}\right)}^i{\left({\displaystyle \frac{s+{\kappa }_1}{s+{\kappa }_2}}-y\right)}^{s-i}.$$

(6)

2. Generalized Operators and Related Lemmas

For any positive integer $\mathsf{ϝ}$, we take the Schurer variant of Bernstein basis function $b_{s,i}^{{\kappa }_1,{\kappa }_2}$ by using shifted knots as follows:

$$b_{s,i,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)=\left({\displaystyle \genfrac{}{}{0pt}{}{s+\mathsf{ϝ}}{i}}\right){\left(y-{\displaystyle \frac{{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}\right)}^i{\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}-y\right)}^{s+\mathsf{ϝ}-i}.$$

(7)

In case of $\mathsf{ϝ}=0$, the Bernstein basis function $b_{s,i}^{{\kappa }_1,{\kappa }_2}$ reduced to $b_{s,i}^{{\kappa }_1,{\kappa }_2}$ by equality (6).

Moreover, the Schurer variant of Bézier bases function ${\tilde{b}}_{s,i}^{{\kappa }_1,{\kappa }_2}$ by means of Bernstein basis function $b_{s,i}^{{\kappa }_1,{\kappa }_2}$ (see [15]) is as follows:

$$\begin{array}{ccc}& & {\tilde{b}}_{s,0,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\lambda ;y)=b_{s,0,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)-{\displaystyle \frac{\lambda }{s+\mathsf{ϝ}+1}}{b}_{s+1,1,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right),\hfill \\ & & {\tilde{b}}_{s,j,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\lambda ;y)=b_{s,j,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)+\lambda ({\displaystyle \frac{s+\mathsf{ϝ}-2j+1}{{(s+\mathsf{ϝ})}^2-1}}{b}_{s+1,j,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)\hfill \\ & & -{\displaystyle \frac{s+\mathsf{ϝ}-2j-1}{{(s+\mathsf{ϝ})}^2-1}}{b}_{s+1,j+1,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)),\mathrm{for}1\le j\le s+\mathsf{ϝ}-1\hfill \\ & & {\tilde{b}}_{s,s,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2,\mathsf{ϝ}}(\lambda ;y)=b_{s,s,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)-{\displaystyle \frac{\lambda }{s+\mathsf{ϝ}+1}}{b}_{s+1,s,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right).\hfill \end{array}$$

Thus, for all ${\displaystyle \frac{{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}\le y\le {\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}$ and real $0\le {\kappa }_1\le {\kappa }_2$, we define the Schurer operators of new $\lambda$-Bernstein shifted knots operators $B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}$ in terms of Bézier bases function ${\tilde{b}}_{s,i,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}$ as follows:

$$B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(g;y)={\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}\sum _{i=0}^{s+\mathsf{ϝ}}{\tilde{b}}_{s,i,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\lambda ;y)g\left({\displaystyle \frac{i}{s}}\right),$$

(8)

where $s\in \mathbb{N}$ (the set of positive integers) and $g\in C[0,1+\mathsf{ϝ}]$ (the collection of all continuously defined functions on $[0,1+\mathsf{ϝ}]$. Özger [15] constructed the Schurer variant of the classical $\lambda$-Bernstein–Bézier bases function, and he defined the Schurer variant of $\lambda$-Bernstein–Bézier bases function ${\tilde{b}}_{s,i,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\lambda ;y)$. In addition, for the choice of Schurer parameter $\mathsf{ϝ}=0$ in ${\tilde{b}}_{s,i,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\lambda ;y)$, the classical $\lambda$-Bernstein–Bézier bases function ${\tilde{b}}_{s,i}(\lambda ;y)$ was obtained by [4]. Other types of operators include the $\lambda$-Schurer–Stancu [25], $\lambda$-Schurer–Kantorovich [26], GBS-type [27] and $\lambda$-Durrmeyer [28].

Remark 1. 

For the operators $B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}$, we obtain the following observations:

1. 

Our new operators clearly reduced to the operators [24] by the choice of $\mathsf{ϝ}=0$ in (8).

2. 

In the equality (8), for the choice ${\kappa }_1={\kappa }_2=0$, our new operators reduced to the operators defined by Özger [15].

3. 

Our new operators minimized to the operators by Cai et al. [3] defined in equality (6) for the choices ${\kappa }_1={\kappa }_2=0$ and $\mathsf{ϝ}=0$ in equality (8).

4. 

For the choice ${\kappa }_1={\kappa }_2=0$, $\mathsf{ϝ}=0$ and $\lambda =0$, our operators were reduced to classical Bernstein operators by [1].

The main format of this paper is the Schurer form of $\lambda$-Bernstein operators using Bézier basis functions. Our goal is to construct the Schurer form of the shifted knots type operators of $\lambda$-Bernstein operators by Bézier bases functions. We use the clasical form of Bernstein shifted knots operators and $\lambda$-Bernstein–Bézier bases functions and then introduce the Schurer form of these operators. We take here the Schure parameter $\mathsf{ϝ}$ and in the case of $\mathsf{ϝ}=0$, we obtain previous operators by [24]. Finally, we compute the convergence by use of Korovkin’s theorem, establish a theorem of local approximation and a theorem of convergence for Lipschitz continuous functions and some direct approximations.

Lemma 1. 

For all $s\in \mathbb{N}\setminus \left\{1\right\}$, if $g\left(\xi \right)=1,\xi ,{\xi }^2$, then the operators $B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}$ defined by (8) have the following equalities:

$$\begin{array}{c}\hfill B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(1;y)=1;\\ \hfill B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\xi ;y)& =& \left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}-{\displaystyle \frac{2\lambda }{s(s+\mathsf{ϝ}-1)}}\right)\left(y-{\displaystyle \frac{{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}\right)\hfill \\ & +& {\displaystyle \frac{\lambda }{s(s+\mathsf{ϝ}-1)}}{\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}{\left(y-{\displaystyle \frac{{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}\right)}^{s+\mathsf{ϝ}+1}\hfill \\ & -& {\displaystyle \frac{\lambda }{s(s+\mathsf{ϝ}-1)}}{\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}{\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}-y\right)}^{s+\mathsf{ϝ}+1}\hfill \\ & +& {\displaystyle \frac{\lambda }{s(s+\mathsf{ϝ}-1)}}\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right);\hfill \end{array}$$

$$\begin{array}{ccc}\hfill B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}({\xi }^2;y)& =& {\left({\displaystyle \frac{s+\mathsf{ϝ}}{s}}\right)}^2[{\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}+{\displaystyle \frac{2\lambda }{s+\mathsf{ϝ}-1}}]\left(y-{\displaystyle \frac{{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}\right)\hfill \\ & +& \left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)[{\displaystyle \frac{s+\mathsf{ϝ}-1}{s}}{\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s}}-{\displaystyle \frac{4\lambda }{s^2}}]{\left(y-{\displaystyle \frac{{\kappa }_1}{s+{\kappa }_2}}\right)}^2\hfill \\ & +& \lambda {\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}[{\displaystyle \frac{{(s+\mathsf{ϝ}+1)}^2}{s^2(s+\mathsf{ϝ}-1)}}+{\displaystyle \frac{1}{s+\mathsf{ϝ}+1}}{\left({\displaystyle \frac{s+\mathsf{ϝ}}{s}}\right)}^2]\hfill \\ & \times & {\left(y-{\displaystyle \frac{{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}\right)}^{s+\mathsf{ϝ}+1}\hfill \\ & +& {\displaystyle \frac{\lambda }{s^2(s+\mathsf{ϝ}-1)}}{\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}{\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}-y\right)}^{s+\mathsf{ϝ}+1}\hfill \\ & & -{\displaystyle \frac{\lambda }{s^2(s+\mathsf{ϝ}-1)}}\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right).\hfill \end{array}$$

Proof. 

We proof the equalities as follows:

$$\begin{array}{ccc}\hfill B_{s,\lambda }^{{\kappa }_1,{\kappa }_2}(1;y)& =& {\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}\sum _{i=0}^{s+\mathsf{ϝ}}{\tilde{b}}_{s,i,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\lambda ;y)\hfill \\ & =& {\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}\{\sum _{i=0}^{s+\mathsf{ϝ}}{b}_{s,i,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)-{\displaystyle \frac{\lambda }{s+\mathsf{ϝ}+1}}{b}_{s+1,1,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)\hfill \\ & +& \lambda {\displaystyle \frac{s+\mathsf{ϝ}-2+1}{{(s+\mathsf{ϝ})}^2-1}}{b}_{s+1,1,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)\hfill \\ & -& \lambda {\displaystyle \frac{s+\mathsf{ϝ}-2-1}{{(s+\mathsf{ϝ})}^2-1}}{b}_{s+1,2,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)+\lambda {\displaystyle \frac{s+\mathsf{ϝ}-4+1}{{(s+\mathsf{ϝ})}^2-1}}{b}_{s+1,2,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)\hfill \\ & -& \lambda {\displaystyle \frac{s+\mathsf{ϝ}-4-1}{{(s+\mathsf{ϝ})}^2-1}}{b}_{s+1,3,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)+\cdots +\lambda {\displaystyle \frac{s+\mathsf{ϝ}-2(s+\mathsf{ϝ}-1)+1}{{(s+\mathsf{ϝ})}^2-1}}{b}_{s+1,s-1,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)\hfill \\ & -& \lambda {\displaystyle \frac{s+\mathsf{ϝ}-2(s+\mathsf{ϝ}-1)-1}{{(s+\mathsf{ϝ})}^2-1}}{b}_{s+1,s,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)-{\displaystyle \frac{\lambda }{s+\mathsf{ϝ}+1}}{b}_{s+1,s,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)\}\hfill \\ & =& {\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}\sum _{i=0}^{s+\mathsf{ϝ}}{b}_{s,i,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)\hfill \\ & =& {\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}{\left(y-{\displaystyle \frac{{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}+{\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}-y\right)}^{s+\mathsf{ϝ}}\hfill \\ & =& 1\hfill \end{array}$$

$$\begin{array}{ccc}\hfill B_{s,\lambda }^{{\kappa }_1,{\kappa }_2}(\xi ;y)& =& {\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}\sum _{i=0}^{s+\mathsf{ϝ}}{\displaystyle \frac{i}{s}}{\tilde{b}}_{s,i}^{{\kappa }_1,{\kappa }_2}(\lambda ;y)\hfill \\ & =& {\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}\left[\sum _{i=0}^{s+\mathsf{ϝ}-1}{\displaystyle \frac{i}{s}}\right\{b_{s,i,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)+\lambda ({\displaystyle \frac{s+\mathsf{ϝ}-2i+1}{{(s+\mathsf{ϝ})}^2-1}}{b}_{s+1,i,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)\hfill \\ & -& {\displaystyle \frac{s+\mathsf{ϝ}-2i-1}{{(s+\mathsf{ϝ})}^2-1}}{b}_{s+1,i+1,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)\left)\right\}+b_{s,s,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)-{\displaystyle \frac{\lambda }{s+\mathsf{ϝ}+1}}{b}_{s+1,s,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)]\hfill \\ & =& {\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}\sum _{i=0}^{s+\mathsf{ϝ}}{\displaystyle \frac{i}{s}}{b}_{s,i,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)\hfill \\ & +& \lambda {\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}\sum _{i=0}^{s+\mathsf{ϝ}}{\displaystyle \frac{i}{s}}{\displaystyle \frac{s+\mathsf{ϝ}-2i+1}{{(s+\mathsf{ϝ})}^2-1}}{b}_{s+1,i,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)\hfill \\ & -& \lambda {\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}\sum _{i=1}^{s+\mathsf{ϝ}-1}{\displaystyle \frac{i}{s}}{\displaystyle \frac{s+\mathsf{ϝ}-2i-1}{{(s+\mathsf{ϝ})}^2-1}}{b}_{s+1,i+1,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right),\hfill \end{array}$$

where the following can be examined:

$$\begin{array}{ccc}\hfill {\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}\sum _{i=0}^{s+\mathsf{ϝ}}{\displaystyle \frac{i}{s}}{b}_{s,i,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)& =& {\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}\left({\displaystyle \frac{s+\mathsf{ϝ}}{s}}\right)\left(y-{\displaystyle \frac{{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}\right)\hfill \\ & \times & \sum _{i=0}^{s+\mathsf{ϝ}-1}{b}_{s-1,i,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)\hfill \\ & =& \left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)\left({\displaystyle \frac{s+\mathsf{ϝ}}{s}}\right)\left(y-{\displaystyle \frac{{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}\right),\hfill \end{array}$$

and

$$\begin{array}{ccc}& & \lambda {\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}\sum _{i=0}^{s+\mathsf{ϝ}}{\displaystyle \frac{i}{s}}{\displaystyle \frac{s+\mathsf{ϝ}-2i+1}{{(s+\mathsf{ϝ})}^2-1}}{b}_{s+1,i,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)\hfill \\ & =& \lambda {\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}[{\displaystyle \frac{1}{s+\mathsf{ϝ}-1}}\sum _{i=0}^{s+\mathsf{ϝ}}{\displaystyle \frac{i}{s}}{b}_{s+1,i,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)-{\displaystyle \frac{2}{{(s+\mathsf{ϝ})}^2-1}}\sum _{i=0}^{s+\mathsf{ϝ}}{\displaystyle \frac{i^2}{s}}{b}_{s+1,i,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)]\hfill \\ & =& \lambda {\displaystyle \frac{s+\mathsf{ϝ}+1}{s(s+\mathsf{ϝ}-1)}}{\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}\left(y-{\displaystyle \frac{{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}\right)\sum _{i=0}^{s+\mathsf{ϝ}-1}{b}_{s,i,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)\hfill \\ & -& \lambda {\displaystyle \frac{2}{s+\mathsf{ϝ}-1}}\left({\displaystyle \frac{s+\mathsf{ϝ}}{s}}\right){\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}{\left(y-{\displaystyle \frac{{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}\right)}^2\sum _{i=0}^{s+\mathsf{ϝ}-2}{b}_{s-1,i,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)\hfill \\ & -& \lambda {\displaystyle \frac{2}{s(s+\mathsf{ϝ}-1)}}{\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}\left(y-{\displaystyle \frac{{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}\right)\sum _{i=0}^{s+\mathsf{ϝ}-1}{b}_{s,i,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)\hfill \\ & =& \lambda {\displaystyle \frac{s+\mathsf{ϝ}+1}{s(s+\mathsf{ϝ}-1)}}{\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}\left(y-{\displaystyle \frac{{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}\right)\hfill \\ & \times & [{\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}-{\left(y-{\displaystyle \frac{{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}\right)}^{s+\mathsf{ϝ}}]\hfill \\ & -& \lambda {\displaystyle \frac{2}{s+\mathsf{ϝ}-1}}\left({\displaystyle \frac{s+\mathsf{ϝ}}{s}}\right){\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}{\left(y-{\displaystyle \frac{{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}\right)}^2\hfill \\ & \times & [{\left({\displaystyle \frac{s+\mathsf{ϝ}}{s+\mathsf{ϝ}+{\kappa }_2}}\right)}^{s+\mathsf{ϝ}-1}-{\left(y-{\displaystyle \frac{{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}\right)}^{s+\mathsf{ϝ}-1}]\hfill \\ & -& \lambda {\displaystyle \frac{2}{s(s+\mathsf{ϝ}-1)}}{\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}\left(y-{\displaystyle \frac{{\kappa }_1}{s+{\kappa }_2}}\right)[{\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}-{\left(y-{\displaystyle \frac{{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}\right)}^{s+\mathsf{ϝ}}]\hfill \\ & =& \lambda {\displaystyle \frac{1}{s}}\left(y-{\displaystyle \frac{{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}\right)-\lambda {\displaystyle \frac{1}{s}}{\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}{\left(y-{\displaystyle \frac{{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}\right)}^{s+\mathsf{ϝ}+1}\hfill \\ & -& \lambda {\displaystyle \frac{2}{s+\mathsf{ϝ}-1}}\left({\displaystyle \frac{s+\mathsf{ϝ}}{s}}\right)\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right){\left(y-{\displaystyle \frac{{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}\right)}^2\hfill \\ & +& \lambda {\displaystyle \frac{2}{s+\mathsf{ϝ}-1}}\left({\displaystyle \frac{s+\mathsf{ϝ}}{s}}\right){\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}{\left(y-{\displaystyle \frac{{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}\right)}^{s+\mathsf{ϝ}+1}.\hfill \end{array}$$

Moreover,

$$\begin{array}{ccc}& -& \lambda {\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}\sum _{i=1}^{s+\mathsf{ϝ}-1}{\displaystyle \frac{i}{s}}{\displaystyle \frac{s+\mathsf{ϝ}-2i+1}{{(s+\mathsf{ϝ})}^2-1}}{b}_{s+1,i+1,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)\hfill \\ & =& -\lambda {\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}{\displaystyle \frac{1}{s}}\left(y-{\displaystyle \frac{{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}\right)\sum _{i=1}^{s+\mathsf{ϝ}-1}{b}_{s,i,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)\hfill \\ & +& \lambda {\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}{\displaystyle \frac{1}{s(s+\mathsf{ϝ}+1)}}\sum _{i=1}^{s+\mathsf{ϝ}-1}{b}_{s+1,i+1,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)\hfill \\ & +& \lambda {\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}{\displaystyle \frac{2}{s+\mathsf{ϝ}-1}}\left({\displaystyle \frac{s+\mathsf{ϝ}}{s}}\right){\left(y-{\displaystyle \frac{{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}\right)}^2\sum _{i=0}^{s+\mathsf{ϝ}-2}{b}_{s-1,i,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)\hfill \\ & -& \lambda {\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}{\displaystyle \frac{2}{s(s+\mathsf{ϝ}-1)}}\left(y-{\displaystyle \frac{{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}\right)\sum _{i=1}^{s+\mathsf{ϝ}-1}{b}_{s,i,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)\hfill \\ & +& \lambda {\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}{\displaystyle \frac{2}{s({(s+\mathsf{ϝ})}^2-1)}}\sum _{i=1}^{s+\mathsf{ϝ}-1}{b}_{s+1,i+1,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)\hfill \\ & =& -\lambda {\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}{\displaystyle \frac{1}{s}}\left(y-{\displaystyle \frac{{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}\right)[{\left({\displaystyle \frac{s+\mathsf{ϝ}}{s+\mathsf{ϝ}+{\kappa }_2}}\right)}^{s+\mathsf{ϝ}}\hfill \\ & -& {\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}-y\right)}^{s+\mathsf{ϝ}}-{\left(y-{\displaystyle \frac{{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}\right)}^{s+\mathsf{ϝ}}]+\lambda {\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}{\displaystyle \frac{1}{s(s+\mathsf{ϝ}+1)}}\hfill \\ & \times & [{\left({\displaystyle \frac{s+\mathsf{ϝ}}{s+\mathsf{ϝ}+{\kappa }_2}}\right)}^{s+\mathsf{ϝ}+1}-{\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}-y\right)}^{s+\mathsf{ϝ}+1}-{\left(y-{\displaystyle \frac{{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}\right)}^{s+\mathsf{ϝ}+1}\hfill \\ & & -(s+\mathsf{ϝ}+1)\left(y-{\displaystyle \frac{{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}\right){\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}-y\right)}^{s+\mathsf{ϝ}}]\hfill \\ & +& \lambda {\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}\left({\displaystyle \frac{s+\mathsf{ϝ}}{s}}\right){\displaystyle \frac{2}{s+\mathsf{ϝ}-1}}{\left(y-{\displaystyle \frac{{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}\right)}^2\hfill \\ & \times & [{\left({\displaystyle \frac{s+\mathsf{ϝ}}{s+\mathsf{ϝ}+{\kappa }_2}}\right)}^{s+\mathsf{ϝ}-1}-{\left(y-{\displaystyle \frac{{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}\right)}^{s+\mathsf{ϝ}-1}]\hfill \\ & -& \lambda {\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}{\displaystyle \frac{2}{s(s+\mathsf{ϝ}-1)}}\left(y-{\displaystyle \frac{{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}\right)[{\left({\displaystyle \frac{s+\mathsf{ϝ}}{s+\mathsf{ϝ}+{\kappa }_2}}\right)}^{s+\mathsf{ϝ}}\hfill \\ & -& {\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}-y\right)}^{s+\mathsf{ϝ}}-{\left(y-{\displaystyle \frac{{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}\right)}^{s+\mathsf{ϝ}}]\hfill \\ & +& \lambda {\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}{\displaystyle \frac{2}{s({(s+\mathsf{ϝ})}^2-1)}}[{\left({\displaystyle \frac{s+\mathsf{ϝ}}{s+\mathsf{ϝ}+{\kappa }_2}}\right)}^{s+\mathsf{ϝ}+1}-{\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}-y\right)}^{s+\mathsf{ϝ}+1}\hfill \\ & -& {\left(y-{\displaystyle \frac{{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}\right)}^{s+\mathsf{ϝ}+1}-(s+\mathsf{ϝ}+1)\left(y-{\displaystyle \frac{{\kappa }_1}{s+{\kappa }_2}}\right){\left({\displaystyle \frac{s+{\kappa }_1}{s+{\kappa }_2}}-y\right)}^s],\hfill \end{array}$$

therefore, we have

$$\begin{array}{ccc}& -& \lambda {\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}\sum _{i=1}^{s+\mathsf{ϝ}-1}{\displaystyle \frac{i}{s}}{\displaystyle \frac{s+\mathsf{ϝ}-2i+1}{{(s+\mathsf{ϝ})}^2-1}}{b}_{s+1,i+1,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)\hfill \\ & =& -\lambda {\displaystyle \frac{s+\mathsf{ϝ}+1}{s(s+\mathsf{ϝ}-1)}}\left(y-{\displaystyle \frac{{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}\right)\hfill \\ & +& \lambda \left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right){\displaystyle \frac{2}{s+\mathsf{ϝ}-1}}\left({\displaystyle \frac{s+\mathsf{ϝ}}{s}}\right){\left(y-{\displaystyle \frac{{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}\right)}^2\hfill \\ & -& \lambda {\displaystyle \frac{1}{s+\mathsf{ϝ}-1}}\left({\displaystyle \frac{s+\mathsf{ϝ}}{s}}\right){\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}{\left(y-{\displaystyle \frac{{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}\right)}^{s+\mathsf{ϝ}+1}\hfill \\ & -& \lambda {\displaystyle \frac{1}{s(s+\mathsf{ϝ}-1)}}{\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}{\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}-y\right)}^{s+\mathsf{ϝ}+1}\hfill \\ & +& \lambda {\displaystyle \frac{1}{s(s+\mathsf{ϝ}-1)}}\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right),\hfill \end{array}$$

this explanation gives $B_{s,\lambda }^{{\kappa }_1,{\kappa }_2,\mathsf{ϝ}}(\xi ;y)$.

Similarly, for $g\left(\xi \right)={\xi }^2$, we find

$$\begin{array}{ccc}\hfill B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}({\xi }^2;y)& =& {\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}\sum _{i=0}^{s+\mathsf{ϝ}}{\displaystyle \frac{i^2}{s^2}}{\tilde{b}}_{s,i,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\lambda ;y)\hfill \\ & =& {\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}\left[\sum _{i=0}^{s+\mathsf{ϝ}-1}{\displaystyle \frac{i^2}{s^2}}\right\{b_{s,i,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)+\lambda ({\displaystyle \frac{s+\mathsf{ϝ}-2i+1}{{(s+\mathsf{ϝ})}^2-1}}{b}_{s+1,i,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)\hfill \\ & -& {\displaystyle \frac{s+\mathsf{ϝ}-2i-1}{{(s+\mathsf{ϝ})}^2-1}}{b}_{s+1,i+1,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)\left)\right\}\hfill \\ & +& {\left({\displaystyle \frac{s+\mathsf{ϝ}}{s}}\right)}^2{b}_{s,s,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)-{\left({\displaystyle \frac{s+\mathsf{ϝ}}{s}}\right)}^2{\displaystyle \frac{\lambda }{s+\mathsf{ϝ}+1}}{b}_{s+1,s,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)]\hfill \\ & =& {\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}\sum _{i=0}^{s+\mathsf{ϝ}}{\displaystyle \frac{i^2}{s^2}}{b}_{s,i,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)\hfill \\ & +& \lambda {\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}\sum _{i=0}^{s+\mathsf{ϝ}}{\displaystyle \frac{i^2}{s^2}}{\displaystyle \frac{s+\mathsf{ϝ}-2i+1}{{(s+\mathsf{ϝ})}^2-1}}{b}_{s+1,i,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)\hfill \\ & -& \lambda {\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}\sum _{i=1}^{s+\mathsf{ϝ}-1}{\displaystyle \frac{i^2}{s^2}}{\displaystyle \frac{s+\mathsf{ϝ}-2i-1}{{(s+\mathsf{ϝ})}^2-1}}{b}_{s+1,i+1,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right).\hfill \end{array}$$

We can easily obtain

$$\begin{array}{ccc}\hfill {\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}\sum _{i=0}^{s+\mathsf{ϝ}}{\displaystyle \frac{i^2}{s^2}}{b}_{s,i,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)& =& {\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^2{\displaystyle \frac{(s+\mathsf{ϝ})(s+\mathsf{ϝ}-1)}{s^2}}{\left(y-{\displaystyle \frac{{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}\right)}^2\hfill \\ & +& {\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}{\displaystyle \frac{(s+\mathsf{ϝ})}{s^2}}\left(y-{\displaystyle \frac{{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}\right),\hfill \end{array}$$

and

$$\begin{array}{ccc}& & \lambda {\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}\sum _{i=0}^{s+\mathsf{ϝ}}{\displaystyle \frac{i^2}{s^2}}{\displaystyle \frac{s+\mathsf{ϝ}-2i+1}{{(s+\mathsf{ϝ})}^2-1}}{b}_{s+1,i,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)\hfill \\ & =& \lambda {\displaystyle \frac{1}{s^2}}\left(y-{\displaystyle \frac{{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}\right)\hfill \\ & +& \lambda \left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right){\displaystyle \frac{(s+\mathsf{ϝ})(s+\mathsf{ϝ}-5)}{s^2(s+\mathsf{ϝ}-1)}}{\left(y-{\displaystyle \frac{{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}\right)}^2\hfill \\ & -& \lambda {\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^2{\displaystyle \frac{2(s+\mathsf{ϝ})}{s^2}}{\left(y-{\displaystyle \frac{{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}\right)}^3\hfill \\ & +& \lambda {\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}{\displaystyle \frac{{(s+\mathsf{ϝ}+1)}^2}{s^2(s+\mathsf{ϝ}-1)}}{\left(y-{\displaystyle \frac{{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}\right)}^{s+\mathsf{ϝ}+1}\hfill \end{array}$$

therefore,

$$\begin{array}{ccc}& & -\lambda {\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}\sum _{i=1}^{s+\mathsf{ϝ}-1}{\displaystyle \frac{i^2}{s^2}}{\displaystyle \frac{s+\mathsf{ϝ}-2i-1}{{(s+\mathsf{ϝ})}^2-1}}{b}_{s+1,i+1,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)\hfill \\ & =& \lambda {\displaystyle \frac{s+\mathsf{ϝ}+1}{s^2(s+\mathsf{ϝ}-1)}}\left(y-{\displaystyle \frac{{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}\right)\hfill \\ & -& \lambda \left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right){\displaystyle \frac{s+\mathsf{ϝ}}{s^2}}{\left(y-{\displaystyle \frac{{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}\right)}^2\hfill \\ & +& \lambda {\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^2{\displaystyle \frac{2(s+\mathsf{ϝ})}{s^2}}{\left(y-{\displaystyle \frac{{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}\right)}^3\hfill \\ & +& \lambda {\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}{\displaystyle \frac{1}{s+\mathsf{ϝ}+1}}{\left({\displaystyle \frac{s+\mathsf{ϝ}}{s}}\right)}^2{\left(y-{\displaystyle \frac{{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}\right)}^{s+\mathsf{ϝ}+1}\hfill \\ & +& \lambda {\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}{\left({\displaystyle \frac{s+\mathsf{ϝ}}{s}}\right)}^2{\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}-y\right)}^{s+\mathsf{ϝ}+1}\hfill \\ & -& \lambda \left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right){\displaystyle \frac{1}{s^2(s+\mathsf{ϝ}-1)}}.\hfill \end{array}$$

Thus, finally, we obtain $B_{s,\lambda }^{{\kappa }_1,{\kappa }_2}({\xi }^2;y)$. □

Lemma 2. 

For the operators $B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}$, we obtain the following central moments of orders one and two:

$$\begin{array}{ccc}\hfill B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\xi -y;y)& =& [\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)-{\displaystyle \frac{2}{s(s+\mathsf{ϝ}-1)}}-1]y\hfill \\ & +& {\displaystyle \frac{1}{s(s+\mathsf{ϝ}-1)}}{\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}\hfill \\ & \times & [{\left(y-{\displaystyle \frac{{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}\right)}^{s+\mathsf{ϝ}+1}-{\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}-y\right)}^{s+\mathsf{ϝ}+1}]\hfill \\ & +& {\displaystyle \frac{1}{s(s+\mathsf{ϝ}-1)}}\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)+{\displaystyle \frac{2{\kappa }_1}{s(s+\mathsf{ϝ}-1)(s+\mathsf{ϝ}+{\kappa }_2)}}-{\displaystyle \frac{{\kappa }_1}{s}};\hfill \end{array}$$

$$\begin{array}{ccc}\hfill B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left({(\xi -y)}^2;y)\right)& =& {\left({\displaystyle \frac{s+\mathsf{ϝ}}{s}}\right)}^2[{\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}+{\displaystyle \frac{2\lambda }{s+\mathsf{ϝ}-1}}]\left(y-{\displaystyle \frac{{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}\right)\hfill \\ & +& \left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)[{\displaystyle \frac{s+\mathsf{ϝ}-1}{s}}{\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s}}-{\displaystyle \frac{4\lambda }{s^2}}]{\left(y-{\displaystyle \frac{{\kappa }_1}{s+{\kappa }_2}}\right)}^2\hfill \\ & +& \lambda {\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}{\left(y-{\displaystyle \frac{{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}\right)}^{s+\mathsf{ϝ}+1}\hfill \\ & \times & [{\displaystyle \frac{{(s+\mathsf{ϝ}+1)}^2}{s^2(s+\mathsf{ϝ}-1)}}+{\displaystyle \frac{1}{s+\mathsf{ϝ}+1}}{\left({\displaystyle \frac{s+\mathsf{ϝ}}{s}}\right)}^2]\hfill \\ & +& {\displaystyle \frac{\lambda }{s^2(s+\mathsf{ϝ}-1)}}{\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}{\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}-y\right)}^{s+\mathsf{ϝ}+1}\hfill \\ & -& {\displaystyle \frac{\lambda }{s^2(s+\mathsf{ϝ}-1)}}\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)\hfill \\ & -& 2y\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}-{\displaystyle \frac{2\lambda }{s(s+\mathsf{ϝ}-1)}}\right)\left(y-{\displaystyle \frac{{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}\right)\hfill \\ & -& {\displaystyle \frac{2\lambda y}{s(s+\mathsf{ϝ}-1)}}{\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}{\left(y-{\displaystyle \frac{{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}\right)}^{s+\mathsf{ϝ}+1}\hfill \\ & +& {\displaystyle \frac{2\lambda y}{s(s+\mathsf{ϝ}-1)}}{\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}{\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}-y\right)}^{s+\mathsf{ϝ}+1}\hfill \\ & -& {\displaystyle \frac{2\lambda y}{s(s+\mathsf{ϝ}-1)}}\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)+y^2.\hfill \end{array}$$

3. Global and Local Approximation

In this section, we derive several global and local approximation theorems for operators (8). We first define the uniform convergence property for our operators using the Ditzian–Totik uniform modulus of smoothness, from which we derive the local and global approximations. We then prove a few simple theorems based on Peetre’s K-functional property and the maximal approximation property of the Lipschitz type. We can substitute a real-valued function that endowed the normed function ${\Vert f\Vert }_{C[0,1]}={sup}_{y\in [0,1]}\left|f\left(y\right)\right|$ for a continuous function f in $C[0,1]$ on $[0,1]$.

Theorem 1 

([29,30]). For every $j=0,1,2$, let $\underset{s\to \infty }{lim}{L}_s\left({\xi }^j;y\right)=y^j$ be a uniform series of positive linear operators on $[a,b]$. Then, $\underset{s\to \infty }{lim}{L}_s\left(f\right)=f$ is uniformly convergent for any compact subset included in $[a,b]$ for each $f\in C[a,b]$.

Theorem 2. 

Considering each f in $C[0,1+\mathsf{ϝ}]$, then we obtain

$$\underset{s\to \infty }{lim}{B}_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(f;y\right)=f\left(y\right)$$

is uniformly convergent on $[0,1]$.

Proof. 

According to Lemma 1, it is evident that for any $j=0,1,2$,

$$\underset{s\to \infty }{lim}{B}_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left({\xi }^j;y\right)=y^j,$$

thus, by applying the famous Bohman–Korovkin–Popoviciu theorem, then $B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(f;y\right)$ uniformly converges to set $f\in C[0,1+\mathsf{ϝ}]$. □

Theorem 3 

([31,32]). Let the operators ${\left\{L_s\right\}}_{s\ge 1}:C[0,1]\to C[0,1]$, such that ${lim}_{s\to \infty }\left|\right|L_s\left({\xi }^j\right)-y^j{\left|\right|}_{C[0,1]}=0,$ for all $j=0,1,2$. Then for any $f\in C[0,1],$ we obtain

$$\underset{s\to \infty }{lim}\left|\right|L_s\left(f\right)-f{\left|\right|}_{C[0,1]}=0.$$

Theorem 4. 

Suppose that $B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}$ assigns as a function $C[0,1+\mathsf{ϝ}]\to C[0,1+\mathsf{ϝ}]$ such that ${lim}_{s\to \infty }\left|\right|B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left({\xi }^j\right)-y^j{\left|\right|}_{C[0,1+\mathsf{ϝ}]}=0$. Then, for all ℓ in $C[0,1+\mathsf{ϝ}]$, we obtain

$$\underset{s\to \infty }{lim}{\Vert B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\mathsf{\ell })-\mathsf{\ell }\Vert }_{C[0,1+\mathsf{ϝ}]}=0.$$

Proof. 

We take in account Theorem 3 and the famous Bohman–Korovkin–Popoviciu theorem, then, easily, we lead to show for all $j=0,1,2$ that

$$\begin{array}{c}\hfill \underset{s\to \infty }{lim}{\parallel B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left({\xi }^j\right)-y^j\parallel }_{C[0,1+\mathsf{ϝ}]}=0.\end{array}$$

According to Lemma 1, it is easy to obtain $\parallel B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}{\left(1\right)-1\parallel }_{C[0,1+\mathsf{ϝ}]}=\underset{y\in [0,1]}{sup}\left|B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(1;y)-1\right|=0$. For $j=1$, it is easy to see

$$\begin{array}{ccc}\hfill \parallel B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}{\left(\xi \right)-y\parallel }_{C[0,1+\mathsf{ϝ}]}& =& \underset{y\in [0,1]}{sup}\left|B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\xi ;y)-y\right|\hfill \\ & \le & \underset{y\in [0,1]}{max}\left|B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\xi -y;y)\right|\hfill \\ & \le & \left|\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)-{\displaystyle \frac{2}{s(s+\mathsf{ϝ}-1)}}-1\right|\hfill \\ & & +\left|{\displaystyle \frac{1}{s(s+\mathsf{ϝ}-1)}}{\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}\right[{\left(1-{\displaystyle \frac{{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}\right)}^{s+\mathsf{ϝ}+1}\hfill \\ & & -{\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}-1\right)}^{s+\mathsf{ϝ}+1}\left]\right|\hfill \\ & & +\left|{\displaystyle \frac{1}{s(s+\mathsf{ϝ}-1)}}\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)+{\displaystyle \frac{2{\kappa }_1}{s(s+\mathsf{ϝ}-1)(s+\mathsf{ϝ}+{\kappa }_2)}}-{\displaystyle \frac{{\kappa }_1}{s}}\right|\hfill \end{array}$$

Since $s\to \infty$ then ${\displaystyle \frac{1}{s+\mathsf{ϝ}+{\kappa }_2}}\to 0,{\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_1}{s+\mathsf{ϝ}+{\kappa }_2}}\to 1,{\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\to 1$; therefore, we obtain $\parallel B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}{\left(\xi \right)-y\parallel }_{C[0,1+\mathsf{ϝ}]}\to 0$. Similarly for $j=2$, we see

$$\begin{array}{ccc}\hfill \parallel B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left({\xi }^2\right)-y^2{\parallel }_{C[0,1+\mathsf{ϝ}]}& =& \underset{y\in [0,1])}{sup}\left|B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}({\xi }^2;y)-y^2\right|,\hfill \end{array}$$

which leads to $\parallel B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left({\xi }^2\right)-y^2{\parallel }_{C[0,1+\mathsf{ϝ}]}\to 0$ as $s\to \infty$. These observations lead to our proof. □

Theorem 5. 

For any $\xi \in C^m[0,1+\mathsf{ϝ}]$, the operators $B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}$ satisfy the following equality:

$$\underset{s\to \infty }{lim}\underset{0\le y\le 1}{sup}{\displaystyle \frac{|B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\xi ;y)-\xi \left(y\right)|}{{(1+y^2)}^{1+k}}}=0,$$

where $C^m[0,1+\mathsf{ϝ})$ denotes the $m^{th}$-order continuously differentiable function on $[0,1+\mathsf{ϝ}]$ and k is a positive number.

Proof. 

If we consider the inequality $\left|\xi \left(y\right)\right|\le (1+y^2)\left|\right|\xi \left|\right|$, we can obtain equality for every real $y_0\in [0,1+\mathsf{ϝ}]$:

$$\begin{array}{ccc}\hfill & & \underset{s\to \infty }{lim}\underset{0\le y\le 1}{sup}{\displaystyle \frac{|B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\xi ;y)-\xi \left(y\right)|}{{(1+y^2)}^{1+k}}}\hfill \\ & & \le \underset{y\le y_0}{sup}{\displaystyle \frac{|B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\xi ;y)-\xi \left(y\right)|}{{(1+y^2)}^{1+k}}}+\underset{y\ge y_0}{sup}{\displaystyle \frac{|B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\xi ;y)-\xi \left(y\right)|}{{(1+y^2)}^{1+k}}}\hfill \\ & & \le \left|\right|B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\xi ;y)-\xi \left(y\right){\left|\right|}_{C[0,y_0]}\hfill \\ & & +\left|\right|\zeta \left|\right|\underset{y\ge y_0}{sup}{\displaystyle \frac{|B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(1+t^2;y)-\xi \left(y\right)|}{{(1+y^2)}^{1+k}}}+\underset{y\ge y_0}{sup}{\displaystyle \frac{\left|\xi \right(y\left)\right|}{{(1+y^2)}^{1+k}}}\hfill \\ & & ={\mathcal{R}}_1+{\mathcal{R}}_2+{\mathcal{R}}_3,\left(\mathrm{we} \mathrm{suppose}\right).\hfill \end{array}$$

Thus, we have

$${\mathcal{R}}_3=\underset{y\ge y_0}{sup}{\displaystyle \frac{\left|\xi \right(y\left)\right|}{{(1+y^2)}^{1+k}}}\le \underset{y\ge y_0}{sup}{\displaystyle \frac{\left|\right|\zeta \left|\right|(1+y^2)}{{(1+y^2)}^{1+k}}}\le {\displaystyle \frac{\left|\right|\xi \left|\right|}{{(1+y_0^2)}^k}}.$$

(9)

Therefore,

$$\underset{s\to \infty }{lim}\underset{y\ge y_0}{sup}{\displaystyle \frac{B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(1+t^2;y)}{1+y^2}}=1.$$

Hence, for any ${ϵ}^{*}>0,$ there are positive integers $s_1$ for every $s\ge s_1$, and we obtain the equality.

$$\underset{y\ge y_0}{sup}{\displaystyle \frac{B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(1+t^2;y)}{1+y^2}}\le {\displaystyle \frac{{(1+y_0^2)}^k}{\left|\right|\xi \left|\right|}}{\displaystyle \frac{{ϵ}^{*}}{3}}+1.$$

For all $s\ge s_1$

$${\mathcal{R}}_2=\left|\right|\zeta \left|\right|\underset{y\ge y_0}{sup}{\displaystyle \frac{B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(1+t^2;y)}{{(1+y^2)}^{1+k}}}\le {\displaystyle \frac{\left|\right|\xi \left|\right|}{{(1+y_0^2)}^k}}+{\displaystyle \frac{{ϵ}^{*}}{3}}.$$

(10)

It is evident to us that (9) and (10) are equal.

$${\mathcal{R}}_2+{\mathcal{R}}_3\le 2{\displaystyle \frac{\left|\right|\xi \left|\right|}{{(1+y_0^2)}^k}}+{\displaystyle \frac{{ϵ}^{*}}{3}}.$$

Selecting any real $y_0$, if it is sufficiently large, then ${\displaystyle \frac{\left|\right|\xi \left|\right|}{{(1+y_0^2)}^k}}\le {\displaystyle \frac{{ϵ}^{*}}{6}}$; thus, we obtain for all $s\ge s_1$

$${\mathcal{R}}_2+{\mathcal{R}}_3\le {\displaystyle \frac{2{ϵ}^{*}}{3}}.$$

(11)

On the other side, if we take $s_2\ge s$, then it is obvious that

$${\mathcal{R}}_1=\left|\right|B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\xi ;y)-\xi \left(y\right){\left|\right|}_{C[0,y_0]}\le {\displaystyle \frac{{ϵ}^{*}}{3}}.$$

(12)

When we combine the equalities (11) and (12) finally, we can easily obtain the outcomes of Theorem 5, which is as follows:

$$\underset{s\to \infty }{lim}\underset{0\le y\le 1}{sup}{\displaystyle \frac{|B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\xi ;y)-\xi \left(y\right)|}{{(1+y^2)}^{1+k}}}<{ϵ}^{*}.$$

□

We provide some results concerning the uniform Ditzian–Totik modulus of smoothness in global approximations. We address the fundamental characteristic of the uniform property of modulus of smoothness for orders one and two, which is

$$\begin{array}{c}\hfill \omega (\phi ,\delta ):=\underset{0<\left|\rho \right|\le \delta }{sup}\underset{y,y+\rho \psi \left(y\right)\in [0,1]}{sup}\left\{\right|\phi (y+\rho \psi \left(y\right))-\phi \left(y\right)\left|\right\};\end{array}$$

$$\begin{array}{c}\hfill {\omega }_2^{\psi }(\phi ,\delta ):=\underset{0<\left|\rho \right|\le \delta }{sup}\underset{y,y\pm \rho \psi \left(y\right)\in [0,1]}{sup}\left\{\right|\phi (y+\rho \psi \left(y\right))-2\phi \left(y\right)+\phi (y-\rho \psi \left(y\right))\left|\right\},\end{array}$$

and the step-weight function $\psi$ on $[0,1]$. Let us consider $\psi$ on $[\alpha ,\beta ]$ such that $\psi \left(y\right)={\left[(y-\alpha )(\beta -y)\right]}^{1/2}$ if $y\in [\alpha ,\beta ]$ (see [33]). If we choose $C^{*}$ to represent the set of all absolutely continuous functions, then the approximate K-functional property of Peetre is as follows:

$$\begin{array}{c}\hfill K_2^{\psi }(\phi ,\delta )=\underset{\zeta \in {\mathsf{ϝ}}^2\left(\psi \right)}{inf}\left\{\delta \left|\right|{\psi }^2{\zeta }^{″}{\left|\right|}_{C[0,1]}+\left|\right|\phi -{\zeta \left|\right|}_{C[0,1]}\right\},\mathrm{for}\mathrm{all}\zeta \in C^2[0,1],\end{array}$$

and for any $\delta >0$, ${\mathsf{ϝ}}^2\left(\psi \right)=\{\zeta \in C[0,1]\mathrm{such}\mathrm{that}{\zeta }^{\prime }\in C^{*}[0,1],{\psi }^2{\zeta }^{″}\in C[0,1]\}$ and $C^2[0,1]=\{\zeta \in C[0,1]:{\zeta }^{\prime },{\zeta }^{″}\in C[0,1]\}$.

Remark 2 

([34]). For a positive real constant M, one has the inequality

$$\begin{array}{c}\hfill M^{-1}{\omega }_2^{\psi }(\phi ,\sqrt{\delta })\le K_2^{\psi }(\phi ,\delta )\le M{\omega }_2^{\psi }(\phi ,\sqrt{\delta }).\end{array}$$

(13)

Theorem 6 

([35]). Provided $C[{\beta }_1,{\beta }_2]\to C[x_1,x_2]$, let ${\left\{R\right\}}_{s\ge 1}$ be any series of operators such that $[{\beta }_1,{\beta }_2]\subseteq [x_1,x_2]$

1. 

Consequently, for all $y\in [x_1,x_2],$ and all ℓ$\in C[{\beta }_1,{\beta }_2]$

$$\begin{array}{ccc}\hfill |R_s(\mathsf{\ell };y)-\mathsf{\ell }\left(y\right)|& \le & |\mathsf{\ell }\left(y\right)||R_s(1;y)-1|\hfill \\ & +& \{R_s(1;y)+{\displaystyle \frac{1}{\delta }}\sqrt{R_s({(\xi -y)}^2;y)}\sqrt{R_s(1;y)}\}\omega (\mathsf{\ell };\delta ),\hfill \end{array}$$

2. 

Thus, for every $y\in [x_1,x_2],$ and every ${\mathsf{\ell }}^{\prime }\in C[{\beta }_1,{\beta }_2]$

$$\begin{array}{ccc}\hfill |K_s(\mathsf{\ell };y)-\mathsf{\ell }\left(y\right)|& \le & |\mathsf{\ell }\left(y\right)||R_s(1;y)-1|+|{\mathsf{\ell }}^{\prime }\left(y\right)\left|\right|R_s(\xi -y;y)|\hfill \\ & +& R_s({(\xi -y)}^2;y)\{\sqrt{R_s(1;y)}+{\displaystyle \frac{1}{\delta }}\sqrt{R_s({(\xi -y)}^2;y)}\}\omega ({\mathsf{\ell }}^{\prime };\delta ).\hfill \end{array}$$

Theorem 7. 

Considering each $\mathsf{\ell }\in C[0,1+\mathsf{ϝ}]$ where $y\in [0,1]$, then $B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}$ defined by (8) yields the following inequality:

$$\begin{array}{ccc}\hfill |B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\mathsf{\ell };y)-\mathsf{\ell }\left(y\right)|& \le & 2\omega \left(\mathsf{\ell };\sqrt{{\delta }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)}\right),\hfill \end{array}$$

where ${\delta }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)=B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}({(\xi -y)}^2;y)$ and ω is a first-order modulus of continuity.

Proof. 

By applying Lemma 1, Lemma 2, and considering (1) from Theorem 6 we obtain

$$\begin{array}{ccc}\hfill |B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\mathsf{\ell };y)-\mathsf{\ell }\left(y\right)|& \le & |\mathsf{\ell }\left(y\right)||B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(1;y)-1|+\{B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(1;y)\hfill \\ & +& {\displaystyle \frac{1}{\delta }}\sqrt{B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}({(\xi -y)}^2;y)}\sqrt{B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(1;y)}\}\omega (\mathsf{\ell };\delta )\hfill \end{array}$$

we suppose $\delta =\sqrt{B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}({(\xi -y)}^2;y)}=\sqrt{{\delta }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)}$, which is the outcome we need. □

Theorem 8. 

If $y\in [0,1]$ and ${\mathsf{\ell }}^{\prime }$ in $C[0,1+\mathsf{ϝ}]$ then operator $B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}$ is followed by

$$\begin{array}{ccc}\hfill |B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\mathsf{\ell };y)-\mathsf{\ell }\left(y\right)|& \le & |{\mathsf{\ell }}^{\prime }\left(y\right)|{\zeta }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)+2{\delta }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)\omega ({\mathsf{\ell }}^{\prime };\sqrt{{\delta }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)}),\hfill \end{array}$$

where ${\zeta }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)={max}_{y\in [0,1]}\left|B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}((\xi -y);y)\right|$ and ${\delta }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)$ specified by Theorem 7.

Proof. 

It is simple to arrive at this conclusion if we take (2) from Theorem 6 and Lemma 1, Lemma 2.

$$\begin{array}{ccc}\hfill |B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\mathsf{\ell };y)-\mathsf{\ell }\left(y\right)|& \le & |\mathsf{\ell }\left(y\right)||B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(1;y)-1|+|{\mathsf{\ell }}^{\prime }\left(y\right)\left|\right|B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\xi -y;y)|\hfill \\ & +& B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}({(\xi -y)}^2;y)\{1+{\displaystyle \frac{\sqrt{B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}({(\xi -y)}^2;y)}}{\delta }}\}\omega ({\mathsf{\ell }}^{\prime };\delta )\hfill \\ & \le & |{\mathsf{\ell }}^{\prime }\left(y\right)|{\zeta }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)+2{\delta }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)\omega ({\mathsf{\ell }}^{\prime };\sqrt{{\delta }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)}),\hfill \end{array}$$

where we let ${\zeta }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)={max}_{y\in [0,1]}\left|B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}((\xi -y);y)\right|$. □

Theorem 9. 

Let $\gamma \left(y\right)$ $(\gamma \ne 0)$ be any step-weight function ${\gamma }^2$ that is concave such that $\gamma \left(y\right)=\left[\right(\beta -y)$${(y-\alpha )]}^{1/2}$ if $y\in [\alpha ,\beta ]$; then, for all $f\in C[0,1+\mathsf{ϝ}]$ and $s\in [0,1]$ operators $B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}$ satisfying

$$\begin{array}{c}\hfill |B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(f;y\right)-f\left(y\right)|\le M{\omega }_2^{\gamma }\left(f,{\displaystyle \frac{{[{\varphi }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)+{\psi }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)]}^{1/2}}{2\gamma \left(y\right)}}\right)+\omega \left(f,{\displaystyle \frac{{\psi }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)}{\gamma \left(y\right)}}\right),\end{array}$$

where ${\psi }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)=B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\xi -y;y)$ and ${\varphi }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)=B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}({(\xi -y)}^2;y).$

Proof. 

Taking into account the following auxiliary operators

$$\begin{array}{c}\hfill {\Omega }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(f;y)=B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(f;y\right)+f\left(y\right)-f\left({\psi }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)+y\right),\end{array}$$

(14)

where $f\in C[0,1+\mathsf{ϝ}],y\in [0,1]$, then by the virtue of Lemma 1, it is easy to obtain the following relations

$${\Omega }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(1;y)=1\mathrm{and}{\Omega }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\xi ;y)=y,$$

${\Omega }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)(\xi -y;y)=0$.

Let $\mathsf{\ell }=\rho y+(1-\rho )t$ for $\rho \in [0,1]$. Following the property ${\gamma }^2\left(y\right)\ge \rho {\gamma }^2\left(y\right)+$$(1-\rho ){\gamma }^2\left(\xi \right)$ as ${\gamma }^2$ is concave on $[0,1]$ and

$$\begin{array}{c}\hfill {\displaystyle \frac{|\xi -\mathsf{\ell }|}{{\gamma }^2\left(y\right)}}\le {\displaystyle \frac{\rho |y-t|}{\rho {\gamma }^2\left(y\right)+(1-\rho ){\gamma }^2\left(\xi \right)}}\le {\displaystyle \frac{|\xi -y|}{{\gamma }^2\left(y\right)}}.\end{array}$$

(15)

We obtain the following identities:

$$\begin{array}{ccc}\hfill |{\Omega }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(f;y)-f\left(y\right)|& & \le |{\Omega }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(f-\zeta ;y)|+|{\Omega }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\zeta ;y)-\zeta \left(y\right)|+|f\left(y\right)-\zeta \left(y\right)|\hfill \\ & & \le {4\parallel f-\zeta \parallel }_{C[0,1]}+|{\Omega }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\zeta ;y)-\zeta \left(y\right)|.\hfill \end{array}$$

By use of Taylor’s series, we can conclude that

$$\begin{array}{ccc}\hfill |{\Omega }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\zeta ;y)-\zeta \left(y\right)|& \le & B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(|{\int }_y^{\xi }|\xi -\mathsf{\ell }||{\zeta }^{″}(\mathsf{\ell })|\mathrm{d}\mathsf{\ell }|;y\right)\hfill \\ \hfill & & +\left|{\int }_y^{{\psi }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)+y}\right|{\psi }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)+y-\mathsf{\ell }|\left|{\zeta }^{″}(\mathsf{\ell })\right|\mathrm{d}\mathsf{\ell }|\hfill \\ \hfill & \le & \parallel {\gamma }^2{\zeta }^{″}{\parallel }_{C[0,1]}{B}_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(|{\int }_y^{\xi }{\displaystyle \frac{|\xi -\mathsf{\ell }|}{{\gamma }^2(\mathsf{\ell })}}\mathrm{d}\mathsf{\ell }|;y\right)+{\parallel {\gamma }^2{\zeta }^{″}\parallel }_{C[0,1]}\hfill \\ \hfill & & \times \left|{\int }_y^{{\psi }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)+y}\right|{\psi }_{s,\lambda }^{{\varkappa }_1,{\varkappa }_2}\left(y\right)+y-\mathsf{\ell }\left|{\displaystyle \frac{\mathrm{d}\mathsf{\ell }}{{\gamma }^2\left(y\right)}}\right|\hfill \\ \hfill & \le & {\gamma }^{-2}\left(y\right){\parallel {\gamma }^2{\zeta }^{″}\parallel }_{C[0,1]}{B}_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}({(\xi -y)}^2;y)\hfill \\ \hfill & & +{\gamma }^{-2}\left(y\right){\psi }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right){\parallel {\gamma }^2{\zeta }^{″}\parallel }_{C[0,1]}.\hfill \end{array}$$

(16)

We use the Peetre’s K-functional properties and the relations (13) and (16); then, it is easy to obtain

$$\begin{array}{cc}\hfill |{\Omega }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)(f;y)-f\left(y\right)|& \le {4\parallel f-\zeta \parallel }_{C[0,1]}+{\gamma }^{-2}\left(y\right){\parallel {\gamma }^2{\zeta }^{″}\parallel }_{C[0,1]}\left({\varphi }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)+{\psi }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)\right)\hfill \\ \hfill & \le M{\omega }_2^{\gamma }\left(f,{\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{{\varphi }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)+{\psi }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)}{\gamma \left(y\right)}}}\right).\hfill \end{array}$$

It is obvious that

$$\begin{array}{cc}\hfill |f\left({\psi }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)+y\right)-f\left(y\right)|& =|f\left({\psi }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)+y\right)-f\left(y\right)|\le \omega \left(f,{\displaystyle \frac{{\psi }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)}{\gamma \left(y\right)}}\right).\hfill \end{array}$$

Thus, finally, we obtain the inequality

$$\begin{array}{cc}\hfill |B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(f;y\right)-f\left(y\right)|& \le |{\Omega }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(f;y)-f\left(y\right)|+|f\left({\psi }_{s,\lambda }^{{\varkappa }_1,{\varkappa }_2}\left(y\right)+y\right)-f\left(y\right)|\hfill \\ \hfill & \le M{\omega }_2^{\gamma }\left(f,{\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{{\varphi }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)+{\psi }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)}{(y-u)(v-y)}}}\right)+\omega \left(f,{\displaystyle \frac{{\psi }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)}{\gamma \left(y\right)}}\right),\hfill \end{array}$$

which completes the desired proof of Theorem 9. □

Theorem 10. 

Assuming $y\in [0,1]$ and $f^{\prime }\in C[0,1+\mathsf{ϝ}]$, then we obtain the following inequality:

$$\begin{array}{cc}\hfill |B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(f;y\right)-f\left(y\right)|& \le {\varphi }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)\left|f^{\prime }\left(y\right)\right|+2\sqrt{{\varphi }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)}\omega \left(f^{\prime },\sqrt{{\varphi }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)}\right).\hfill \end{array}$$

Proof. 

We are aware of the relationship

$$\begin{array}{c}\hfill f\left(\xi \right)=f\left(y\right)+f^{\prime }\left(y\right)(\xi -y)+{\int }_y^{\xi }(f^{\prime }\left(z\right)-f^{\prime }\left(y\right))\mathrm{d}z,\end{array}$$

(17)

for all $t,y\in [0,1]$. Upon applying the operators $B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}$ to equality (17), we obtain

$$\begin{array}{c}\hfill B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(f\left(\xi \right)-f\left(y\right);y)=f^{\prime }\left(y\right)B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\xi -y;y)+B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left({\int }_y^{\xi }(f^{\prime }\left(z\right)-f^{\prime }\left(y\right))\mathrm{d}z;y\right).\end{array}$$

Given any $f\in C[0,1+\mathsf{ϝ}]$, when $y\in [0,1]$, then one possesses

$$|f\left(\xi \right)-f\left(y\right)|\le \left(1+{\displaystyle \frac{|\xi -y|}{\delta }}\right)\omega (f,\delta ),\delta >0.$$

Given the inequality above, we have

$$\left|{\int }_{\xi }^y(f^{\prime }\left(y\right)-f^{\prime }\left(z\right))\mathrm{d}z\right|\le \left(|\xi -y|+{\displaystyle \frac{{(\xi -y)}^2}{\delta }}\right)\omega (f^{\prime },\delta ).$$

Therefore, it is easy to obtain

$$\begin{array}{cc}\hfill |B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(f;y\right)-f\left(y\right)|& \le |f^{\prime }\left(y\right)\left|\right|B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\xi -y;y)|\hfill \\ \hfill & +\omega (f^{\prime },\delta )\left\{{\displaystyle \frac{1}{\delta }}{B}_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}({(\xi -y)}^2;y)+B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(\right|\xi -y|;y)\right\}.\hfill \end{array}$$

The Cauchy–Schwarz inequality yields

$$B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(\right|\xi -y|;y)\le B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}{(1;y)}^{{\displaystyle \frac{1}{2}}}{B}_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}{({(\xi -y)}^2;y)}^{{\displaystyle \frac{1}{2}}}=B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}{({(\xi -y)}^2;y)}^{{\displaystyle \frac{1}{2}}}.$$

Thus, we have

$$\begin{array}{cc}\hfill |B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(f;y\right)-f\left(y\right)|& \le f^{\prime }\left(y\right)B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}({(\xi -y)}^2;y)\hfill \\ \hfill & +\left\{{\displaystyle \frac{1}{\delta }}\sqrt{B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}({(\xi -y)}^2;y)}+1\right\}{B}_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}{({(\xi -y)}^2;y)}^{{\displaystyle \frac{1}{2}}}\omega (f^{\prime },\delta ),\hfill \end{array}$$

by use of $\delta =\sqrt{B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}({(\xi -y)}^2;y)}$, then we obtain the desired results. □

Next, recalling [36], we estimate the local direct approximation by means of a Lipschitz-type maximum function.

Theorem 11. 

Let $f\in Li{p}_K\left(\kappa \right):=\{f\in C[0,1+\mathsf{ϝ}]suchthat|f\left(\xi \right)-f\left(y\right)|\le K{\displaystyle \frac{{|\xi -y|}^{\kappa }}{{({\xi }_1{y}^2+{\xi }_{2}y+t)}^{\frac{\kappa }{2}}}};$$y,t\in [0,1]\}$. Then, for any $\kappa \in (0,1]$, we have

$$\begin{array}{cc}\hfill |B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(f;y\right)-f\left(y\right)|& \le K\sqrt{{\displaystyle \frac{{\left[{\varphi }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)\right]}^{\kappa }}{{({\xi }_1{y}^2+{\xi }_{2}y)}^{\kappa }}}},\hfill \end{array}$$

where ${\xi }_1\ge 0,{\xi }_2>0,\kappa \in (0,1]$ and $K>0$ for any constant (see [36]).

Proof. 

For every $\kappa \in (0,1]$, consider $f\in Li{p}_K\left(\kappa \right)$. Initially, we aim to demonstrate the validity of our result for $\kappa =1$. Thus, we obtain for any $f\in Li{p}_K\left(1\right)$

$$\begin{array}{cc}\hfill |B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(f;y\right)-f\left(y\right)|& \le |B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(\right|f\left(\xi \right)-f\left(y\right)|;y)|+f\left(y\right)|B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(1;y)-1|\hfill \\ \hfill & \le {\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}\sum _{i=0}^{s+\mathsf{ϝ}}{\tilde{b}}_{s,i,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\lambda ;y)|f\left(\xi \right)-f\left(y\right)|\hfill \\ \hfill & \le K{\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}\sum _{i=0}^{s+\mathsf{ϝ}}{\displaystyle \frac{{\tilde{b}}_{s,i,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\lambda ;y)|\xi -y|}{{({\xi }_1{y}^2+{\xi }_{2}y+t)}^{\frac{1}{2}}}}.\hfill \end{array}$$

By using

$${({\xi }_1{y}^2+{\xi }_{2}y+t)}^{-1/2}\le {({\xi }_1{y}^2+{\xi }_{2}y)}^{-1/2}({\xi }_1\ge 0,{\xi }_2>0)$$

and by the Cauchy–Schwarz inequality, we see

$$\begin{array}{cc}\hfill |S_{s,\lambda }^{{\varkappa }_1,{\varkappa }_2}\left(f;y\right)-f\left(y\right)|& \le K{\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}{({\xi }_1{y}^2+{\xi }_{2}y)}^{-1/2}\sum _{i=0}^{s+\mathsf{ϝ}}{\tilde{b}}_{s,i,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\lambda ;y)|\xi -y|\hfill \\ \hfill & =K{({\xi }_1{y}^2+{\xi }_{2}y)}^{-1/2}\left|B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\xi -y;y)\right|\hfill \\ \hfill & \le K{({\xi }_1{y}^2+{\xi }_{2}y)}^{-1/2}{B}_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(\right|\xi -y|;y)\hfill \\ \hfill & \le K{\left(B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}({(\xi -y)}^2;y)\right)}^{{\displaystyle \frac{1}{2}}}{({\xi }_1{y}^2+{\xi }_{2}y)}^{-1/2}.\hfill \end{array}$$

Our result is therefore true for $\kappa =1$. Furthermore, by introducing the Hölder’s inequality and applying the monotonicity property to operators $B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}$ for $\kappa \in (0,1]$, the following requirements will also be satisfied, thus

$$\begin{array}{ccc}\hfill \left|B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(f;y\right)-f\left(y\right)\right|& \le & {\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}\sum _{i=0}^{s+\mathsf{ϝ}}{\tilde{b}}_{s,i,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\lambda ;y)|f\left(\xi \right)-f\left(y\right)|\hfill \\ & \le & {\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}\sum _{i=0}^{s+\mathsf{ϝ}}{\tilde{b}}_{s,i,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\lambda ;y)|f\left(\xi \right)-f\left(y\right)|{)}^{{\displaystyle \frac{\kappa }{2}}}{\left(B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(1;y)\right)}^{{\displaystyle \frac{2-\kappa }{2}}}\hfill \\ & \le & K{\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}\sum _{i=0}^{s+\mathsf{ϝ}}{\displaystyle \frac{{\tilde{b}}_{s,i,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\lambda ;y){\left(\xi -y\right)}^2}{{\xi }_1{y}^2+{\xi }_{2}y+t}}{)}^{{\displaystyle \frac{\kappa }{2}}}\hfill \\ & \le & K{\left\{\sum _{i=0}^{s+\mathsf{ϝ}}{\tilde{b}}_{s,i,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\lambda ;y){\left(\xi -y\right)}^2\right\}}^{{\displaystyle \frac{\kappa }{2}}}{({\xi }_1{y}^2+{\xi }_{2}y+t)}^{-\kappa /2}\hfill \\ & \le & K{\left({\displaystyle \frac{s+\mathsf{ϝ}+{\kappa }_2}{s+\mathsf{ϝ}}}\right)}^{s+\mathsf{ϝ}}{({\xi }_1{y}^2+{\xi }_{2}y)}^{-\kappa /2}{\left[B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}({(\xi -y)}^2;y)\right]}^{{\displaystyle \frac{\kappa }{2}}}\hfill \\ & =& K\sqrt{{\displaystyle \frac{{\left[{\varphi }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)\right]}^{\kappa }}{{({\xi }_1{y}^2+{\xi }_{2}y)}^{\kappa }}}}.\hfill \end{array}$$

□

Here, we also prove another local approximation property for the operators of $B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}$ by using the Lipschitz maximal function. Suppose $\Psi$ is the maximal functions of Lipschitz type such that $\Psi \in C_B[0,1]$. For all $t,y\in [0,1]$ the class of Lipschitz-type functions $\Psi$ is defined by (see [37])

$${\omega }_{\vartheta }(\Psi ;y)=\underset{t\ne y,t\in [0,1]}{sup}{\displaystyle \frac{\mid \Psi \left(\xi \right)-\Psi \left(y\right)\mid }{\mid \xi -y{\mid }^{\vartheta }}},$$

(18)

where $0<\vartheta \le 1$.

Theorem 12. 

Considering $\Psi \in C_B[0,1+\mathsf{ϝ}]$ and $y\in [0,1]$, then operators $B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}$ fulfill

$$\left|B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\Psi ;y)-\Psi \left(y\right)\right|\le {\left({\delta }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)\right)}^{{\displaystyle \frac{\vartheta }{2}}}{\omega }_{\vartheta }(\Psi ;y),$$

where ${\delta }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)$ is obtained by Theorem 7 and ${\omega }_{\vartheta }(\Psi ;y)$ is defined by (18). Moreover, $C_B[0,1+\mathsf{ϝ}]$ denotes the set of all continuously bounded functions on $[0,1+\mathsf{ϝ}]$.

Proof. 

By using the Hölder’s inequality, we obtain

$$\begin{array}{ccc}\hfill \left|B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\Psi ;y)-\Psi \left(y\right)\right|& \le & B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(\left|\Psi \left(\xi \right)-\Psi \left(y\right)\right|;y\right)\hfill \\ & \le & {\omega }_{\vartheta }(\Psi ;y)\mid B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left({\left|\xi -y\right|}^{\vartheta };y\right)\hfill \\ & \le & {\omega }_{\vartheta }(\Psi ;y){\left(B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(1;y)\right)}^{{\displaystyle \frac{2-\vartheta }{2}}}{\left(B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}({\left|\xi -y\right|}^2;y)\right)}^{{\displaystyle \frac{\vartheta }{2}}}\hfill \\ & =& {\left(B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left({(\xi -y)}^2;y\right)\right)}^{{\displaystyle \frac{\vartheta }{2}}}{\omega }_{\vartheta }(\Psi ;y).\hfill \end{array}$$

This completes the proof. □

4. Directs Theorem of Operators ${\mathit{B}}_{\mathit{s},\mathit{\lambda },\mathsf{ϝ}}^{{\mathit{\kappa }}_{\mathbf{1}},{\mathit{\kappa }}_{\mathbf{2}}}$

For our new operators $B_{s,\lambda }^{{\kappa }_1,{\kappa }_2}$ by 8 in the space of Peetre’s K-functional, this section can offer some direct approximation results. For every $\Psi \in C[0,1]$ and any $\delta >0$, the basic idea of Peetre’s K-functional $K_p(\Psi ;\delta )$ is defined as follows:

$$K_{\Psi }(\Psi ;\delta )=inf\left\{\left(\delta \Vert {\mathsf{\ell }}^{\prime \prime }{\Vert }_{C[0,1]}\right)+{\Vert \Psi -\mathsf{\ell }\Vert }_{C[0,1]}:\mathsf{\ell },{\mathsf{\ell }}^{\prime },{\mathsf{\ell }}^{″}\in C[0,1]\right\}$$

(19)

For a positive absolute constant M, we have [34] that

$$K_{\Psi }(\Psi ;\delta )\le M{\omega }_{\Psi }(\Psi ;\sqrt{\delta }),\delta >0,$$

$$K_{\Psi }(\Psi ;\delta )\le C\{{\omega }_{\Psi }(\Psi ;\sqrt{\delta })+min(1,\delta ){\left|\right|\Psi \left|\right|}_{C[0,1]}\},$$

where the modulus of continuity for the second order is determined by ${\omega }_{\Psi }(\Psi ;\delta )$, which can be expressed as follows:

$${\omega }_{\Psi }(\Psi ;\delta )=\underset{0<\nu <\delta }{sup}\underset{y\in [0,1])}{sup}|\Psi \left(y\right)+\Psi (y+2\nu )-2\Psi (y+\nu )|.$$

(20)

Theorem 13. 

Let us define an auxiliary operator $A_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}$ for an arbitrary $\Psi \in C[0,1+\mathsf{ϝ}]$, such that

$$A_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\Psi ;y)=B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\Psi ;y)+\Psi \left(y\right)-\Psi \left(B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\Phi ;y)\right),$$

(21)

subsequently, for each $\Phi \in C[0,1+\mathsf{ϝ}]$ we obtain that

$$\begin{array}{ccc}\hfill \left|B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\Phi ;y)-\Phi \left(y\right)\right|& \le & C{\omega }_{\Phi }(\Phi ;{\displaystyle \frac{\sqrt{{\varphi }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)+({\psi }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right){)}^2}}{2}})+{\omega }_{\Phi }(\Phi ;{\psi }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right))\hfill \end{array}$$

where ${\psi }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)=B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left((\xi -y);y\right)$ and ${\varphi }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)=B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left({(\xi -y)}^2;y\right)$.

Proof. 

It is simple to prove that $A_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(1;y)=1$ and

$$A_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\xi ;y)=B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\xi ;y)=y.$$

The Taylor series statement allows us to infer the equality:

$$\mathcal{L}\left(\xi \right)=\mathcal{L}\left(y\right)+(\xi -y){\mathcal{L}}^{\prime }\left(y\right)+{\int }_y^{\xi }(\xi -\mathcal{W}){\mathcal{L}}^{″}\left(\mathcal{W}\right)\mathrm{d}\mathcal{W},\mathcal{L}\in C^2[0,1].$$

(22)

Apply $A_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}$ then

$$\begin{array}{ccc}\hfill A_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\mathcal{L};y)-\mathcal{L}\left(y\right)& =& {\mathcal{L}}^{\prime }\left(y\right)A_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\xi -y;y)+A_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left({\int }_y^{\xi }(\xi -\mathcal{W}){\mathcal{L}}^{\prime \prime }\left(\mathcal{W}\right)\mathrm{d}\mathcal{W};y\right)\hfill \\ & =& A_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left({\int }_y^{\xi }(\xi -\mathcal{W}){\mathcal{L}}^{\prime \prime }\left(\mathcal{W}\right)\mathrm{d}\mathcal{W};y\right)\hfill \\ & =& B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left({\int }_y^{\xi }(\xi -\mathcal{W}){\mathcal{L}}^{\prime \prime }\left(\mathcal{W}\right)\mathrm{d}\mathcal{W};y\right)+{\int }_y^y(y-\mathcal{W}){\mathcal{L}}^{\prime \prime }\left(\mathcal{W}\right)\mathrm{d}\mathcal{W};y\hfill \\ & -& {\int }_y^{B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\xi ;y)}(B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\xi ;y)-\mathcal{W}){\mathcal{L}}^{\prime \prime }\left(\mathcal{W}\right)\mathrm{d}\mathcal{W}\hfill \end{array}$$

$$\begin{array}{ccc}\hfill \mid A_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\mathcal{L};y)-\mathcal{L}\left(y\right)\mid & \le & \left|B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left({\int }_y^{\xi }(\xi -\mathcal{W}){\mathcal{L}}^{\prime \prime }\left(\mathcal{W}\right)\mathrm{d}\mathcal{W};y\right)\right|\hfill \\ & +& \left|{\int }_y^{B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\xi ;y)}\left(B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\xi ;y)-\mathcal{W}\right){\mathcal{L}}^{\prime \prime }\left(\mathcal{W}\right)\mathrm{d}\mathcal{W}\right|.\hfill \end{array}$$

We know the inequality

$$\left|{\int }_y^{\xi }(\xi -\mathcal{W}){\mathcal{L}}^{″}\left(\mathcal{W}\right)\mathrm{d}\mathcal{W}\right|\le {(\xi -y)}^2\Vert {\mathcal{L}}^{\prime \prime }\Vert$$

and

$$\left|{\int }_y^{B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\xi ;y)}\left(B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\xi ;y)-\mathcal{W}\right){\mathcal{L}}^{″}\left(\mathcal{W}\right)\mathrm{d}\mathcal{W}\right|\le (B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\xi ;y)-y{)}^2\Vert {\mathcal{L}}^{\prime \prime }\Vert .$$

Thus, we obtain

$$\mid A_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\mathcal{L};y)-\mathcal{L}\left(y\right)\mid \le \{B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left({(\xi -y)}^2;y\right)+(B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\xi ;y)-y{)}^2\}\Vert {\mathcal{L}}^{\prime \prime }\Vert .$$

On the other hand, we deduce that

$$\Vert B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\Psi ;y)\Vert \le \Vert \Psi \Vert ,$$

and for any $\Psi \in C[0,1+\mathsf{ϝ}]$, we have

$$\mid A_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\Psi ;y)\mid \le \mid B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\Psi ;y)\mid +\mid \Psi \left(y\right)\mid +|\Psi \{B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\Psi ;y)\left\}\right|\le 3\Vert \Psi \Vert ,$$

(23)

By accounting for (22) and (23), we arrive at

$$\begin{array}{ccc}\hfill \left|B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\Phi ;y)-\Phi \left(y\right)\right|& \le & |A_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\Phi -\mathcal{L};y)-(\Phi -\mathcal{L})\left(y\right)|\hfill \\ & +& |A_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\mathcal{L};y)-\mathcal{L}\left(y\right)|+|\Phi \left(y\right)-\Phi \left(B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\xi ;y)\right)|\hfill \\ & \le & 4\left|\left|\Phi -\mathcal{L}\right|\right|+{\omega }_{\Phi }(\Phi ;B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left((\xi -y);y\right))\hfill \\ & +& \{B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left({(\xi -y)}^2;y\right)+\Vert {\mathcal{L}}^{\prime \prime }\Vert (B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\xi -y;y){)}^2\}.\hfill \end{array}$$

Taking infimum over all $\mathcal{L}\in C^2[0,1]$ and using Peetre’s K-functional properties, we obtain

$$\begin{array}{ccc}\hfill \left|B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}(\Phi ;y)-\Phi \left(y\right)\right|& \le & 4K_{\Phi }(\Phi ;{\displaystyle \frac{{\delta }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)+(B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left((\xi -y);y\right){)}^2}{4}}\hfill \\ & & +{\omega }_{\Phi }(\Phi ;B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left((\xi -y);y\right))\hfill \\ & \le & C{\omega }_{\Phi }(\Phi ;{\displaystyle \frac{\sqrt{{\delta }_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(y\right)+(B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left((\xi -y);y\right){)}^2}}{2}})\hfill \\ & & +{\omega }_{\Phi }(\Phi ;B_{s,\lambda ,\mathsf{ϝ}}^{{\kappa }_1,{\kappa }_2}\left(\xi -y;y\right)).\hfill \end{array}$$

Thus, we have our desired proof. □

5. Conclusions and Observation

This paper clearly concludes that the operators (8) are defined as the Schurer parameter of shifted knots operators by the Bézier bases of $\lambda$–Bernstein operators [24]. The operators in (8) reduced to [24] if $\mathsf{ϝ}=0$. In the event that ${\kappa }_1={\kappa }_2=0$, operators (8) reduced to [15]. Considering ${\kappa }_1={\kappa }_2=0$, $\mathsf{ϝ}=0$, then (8) reduced to [3]. Also, ${\kappa }_1={\kappa }_2=0$, when $\mathsf{ϝ}=0$ and $\lambda =0$, the operators (8) are simplified to classical Bernstein operators as shown in [1]. Since our operators (8) are expanded variants of the operators described by [1,3,15,24], in conclusion, we can say that these operators represent a particular case of our new operators (8). For future work, one can obtain the approximation of bivariate-type and q-integer-type operators of (8).