Approximation Properties Variant of Baskakov–Schurer–Szász Operators Induced by Sheffer Polynomials

Abstract

This paper introduces a new class of Baskakov–Schurer–Szász operators generated by Sheffer polynomials in a special case. We establish a Korovkin-type theorem for these operators and investigate their uniform convergence and error estimates using the Ditzian–Totik modulus of smoothness. We also derive weighted approximation and shape-preserving properties for this class of operators. These results extend the theory of positive linear operators and provide tools applicable in approximation and numerical analysis.

1. Introduction

One of the foundational results in approximation theory is the Weierstrass Approximation Theorem [1], which asserts that any continuous function $f\left(x\right)$ defined on a closed interval $[a,b]$ can be uniformly approximated by an algebraic polynomial $P\left(x\right)$ with real coefficients at every point $x\in [a,b]$.

The aim of concrete algebraic functions that provide effective approximation has been a central topic in approximation theory. Among the various approaches, polynomial operators have been extensively employed. In this context, one of the most studied families of operators in recent years is the sequence of Szász operators and their extensions/generalizations (for example, see [2,3,4,5,6,7,8,9]). The classical Szász operators are defined by the following (see [10]):

$$S_n(f,x)=e^{-nx}{\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{{\left(nx\right)}^k}{k!}}f\left({\displaystyle \frac{k}{n}}\right),x\in [0,\infty ).$$

Several generalizations of these operators have been considered. A notable one was introduced by Jakimovski and Leviatan [11], who defined a generalized version using Appell polynomials as follows:

$$P_n(f,x)={\displaystyle \frac{e^{-nx}}{g\left(1\right)}}{\displaystyle \sum _{k=0}^{\infty }}{p}_k\left(nx\right)f\left({\displaystyle \frac{k}{n}}\right),$$

where $p_k\left(x\right)$, for $k\ge 0$, are the Appell polynomials defined via the generating function

$$g\left(t\right)e^{xt}={\displaystyle \sum _{k=0}^{\infty }}{p}_k\left(x\right){\displaystyle \frac{t^k}{k!}},\mathrm{with}g\left(t\right)={\displaystyle \sum _{k=0}^{\infty }}{a}_k{\displaystyle \frac{t^k}{k!}}.$$

Here, $g\left(t\right)$ is assumed to be analytic within the disk $\left|t\right|<R$ for some $R>1$, and $g\left(1\right)\ne 0$.

A further generalization was introduced by Ismail [12] using Sheffer-type polynomials. The corresponding operators are given by the following:

$$T_n(f,x)={\displaystyle \frac{e^{-nxH\left(1\right)}}{g\left(1\right)}}{\displaystyle \sum _{k=0}^{\infty }}{s}_k\left(nx\right)f\left({\displaystyle \frac{k}{n}}\right),$$

where the Sheffer polynomials $s_k\left(x\right)$ are defined via the generating function

$$g\left(t\right)e^{xH\left(t\right)}={\displaystyle \sum _{k=0}^{\infty }}{s}_k\left(x\right){\displaystyle \frac{t^k}{k!}}.$$

In this formulation, $g\left(t\right)$ is as before, and $H\left(t\right)={\sum }_{k=0}^{\infty }{h}_k{\displaystyle \frac{t^k}{k!}}$ is another analytic function in the disk $\left|t\right|<R$ with $R>1$, satisfying the conditions $g\left(1\right)\ne 0$ and $H^{\prime }\left(1\right)=1$. In this paper, we restrict ourselves to the case where $g\left(t\right)=g\left(1\right)>0$ is constant and $H\left(t\right)=H\left(1\right)t$ with $H\left(1\right)>0$. Consequently, the associated Sheffer polynomials reduce to the following:

$$s_k\left(x\right)=g\left(1\right){\left(H\left(1\right)\right)}^k{x}^k.$$

The aim of this paper is to propose a new class of Baskakov–Schurer–Szász operators generated by Sheffer polynomials in the particular case where $s_k\left(x\right)=g\left(1\right){\left(H\left(1\right)\right)}^k{x}^k$. These Baskakov–Schurer–Szász operators are important since they generalize the classical Baskakov [13], Schurer, and Szász [10] operators, as well as the Baskakov–Schurer–Szász operators (see [3,14]). These generalized operators provide another perspective in approximation theory and introduce new classes of operators with richer approximation properties (see [2]). Moreover, they allow the extension of several known approximation results and open new directions for the study of approximation processes, including applications to numerical methods for differential and partial differential equations.

The paper is organized as follows: In Section 2, we construct a new class of Baskakov–Schurer–Szász operators generated by Sheffer polynomials (see (2)) and present the first moments and central moments of these operators. In Section 3, we study their uniform convergence and error estimates using the Ditzian–Totik modulus of smoothness. Then, in Section 4, we derive the weighted approximation and shape-preserving properties for this new class of operators.

2. Construction of the Operators and Preliminary Results

The Baskakov–Schurer–Szász operators are defined as follows:

$$L_{m,q}(w,z)=(m+q){\displaystyle \sum _{j=0}^{\infty }}{c}_{mq}^j\left(z\right){\int }_0^{\infty }w\left(u\right)b_{mq}^j\left(u\right)du,$$

(1)

where $c_{mq}^j\left(z\right)=({\displaystyle \genfrac{}{}{0pt}{}{m+q+j-1}{j}}){\displaystyle \frac{z^j}{{(1+z)}^{m+q+j}}}$ and $b_{mq}^j\left(u\right)=e^{-(m+q)u}{\displaystyle \frac{{\left(\right(m+q\left)u\right)}^j}{j!}}.$

For $z\in [0,\infty )$, we define a new class of Baskakov–Schurer–Szász operators induced by Sheffer polynomials by the following:

$$N_{m,q}(w,z)=(m+q){\displaystyle \sum _{j=0}^{\infty }}{c}_{mq}^j\left(z\right){\int }_0^{\infty }{\displaystyle \frac{w\left(u\right)b_{mq}^j\left(u\right)}{H^j\left(1\right)g\left(1\right)}}du,$$

(2)

where

$$c_{mq}^j\left(z\right)=\left({\displaystyle \genfrac{}{}{0pt}{}{m+q+j-1}{j}}\right){\displaystyle \frac{z^j}{{(1+z)}^{m+q+j}}},$$

and

$$\begin{array}{c}\hfill b_{mq}^j\left(u\right)=e^{-(m+q)u}{\displaystyle \frac{s_j\left((m+q)u\right)}{j!}}=e^{-(m+q)u}{\displaystyle \frac{{\left((m+q)u\right)}^j{\left[H\left(1\right)\right]}^{j}g\left(1\right)}{j!}},\end{array}$$

(3)

with $g\left(1\right),H\left(1\right)>0$, $m\ge 1$, and $q>0$. Clearly, the operators $N_{m,q}$ are positive linear operators, that is, $N_{m,q}(w,z)\ge 0$ whenever $w\left(z\right)\ge 0$. The main advantage of the proposed operators is that they extend the classical Baskakov [13], Baskakov–Szász, and Baskakov–Schurer–Szász operators (see [14]) by incorporating Sheffer polynomials, thereby providing a more general and flexible approximation framework. As a result, several known results can be unified and generalized, and new approximation properties on unbounded intervals can be obtained (see [2]). Compared with existing operators, the proposed class offers greater generality and potential for improved approximation behavior. In particular, such positive linear operators are useful in the approximation of solutions of differential and partial differential equations, for example, in semigroup approaches to the heat equation.

In the sequel, we will find moments of the operators (2) and central moments. Before deriving the moments and central moments, we note that the presence of the Sheffer polynomial structure in the Baskakov–Schurer–Szász operators leads to more involved expressions than in the classical cases, requiring careful manipulation of the generating functions.

Proposition 1.

Let $e_k\left(t\right)=t^k.$ For $k=0,1,\dots ,10,$

$$N_{m,q}(e_k,z)={\displaystyle \frac{m!}{{(m+q)}^k}}+{\displaystyle \frac{1}{{(m+q)}^k}}{\displaystyle \sum _{j=1}^k}(k-j)!{\left({\displaystyle \genfrac{}{}{0pt}{}{k}{j}}\right)}^2{z}^k{\displaystyle \prod _{i=0}^{k-1}}(m+q+i).$$

Proof. 

By (3), we have that $s_j\left((m+q)u\right)={\left((m+q)u\right)}^j{\left[H\left(1\right)\right]}^{j}g\left(1\right)$. Thus, by (2), we have

$$N_{m,q}(w,z)=(m+q){\displaystyle \sum _{j=0}^{\infty }}{c}_{mq}^j\left(z\right){\int }_0^{\infty }{\displaystyle \frac{w\left(u\right)e^{-(m+q)u}{\left((m+q)u\right)}^j}{j!}}du.$$

Define $a_{k,j}={\int }_0^{\infty }{\displaystyle \frac{e^{-(m+q)u}{u}^{k+j}}{j!}}$. Then

$$a_{k,j}={\left.{\displaystyle \frac{-1}{m+q}}{\displaystyle \frac{e^{-(m+q)u}{u}^{k+j}}{j!}}\right|}_{u=0}^{\infty }+{\displaystyle \frac{1}{j(m+q)}}{a}_{k,j-1}={\displaystyle \frac{1}{j(m+q)}}{a}_{k,j-1}$$

with $a_{k,0}={\displaystyle \frac{1}{m+q}}$. By induction on $k,j$, we have

$$a_{k,j}={\displaystyle \frac{(k+j)!}{j!{(m+q)}^{2j+1}}},$$

which implies

$$N_{m,q}(e_k,z)={\displaystyle \sum _{j=0}^{\infty }}\left({\displaystyle \genfrac{}{}{0pt}{}{m+q+j-1}{j}}\right){\displaystyle \frac{(k+j)!z^j}{j!{(m+q)}^j{(1+z)}^{m+q+j}}}.$$

Thus, by direct calculations (symbolic computation) for $k=0,1,\dots ,10$, we complete the proof. □

Note that Proposition 1 has been verified explicitly for $m=0,1,\dots ,10$ using symbolic computation (Maple). The general case remains open, since we did not succeed in proving the closed-form expression for the above sum for arbitrary m. The restriction $m=10$ is computational due to the complexity of expressions for general m.

Example 1.

For instance, Proposition 1 gives

$$\begin{array}{cc}\hfill N_{m,q}(e_0,z)& =1,\hfill \\ \hfill N_{m,q}(e_1,z)& ={\displaystyle \frac{1}{m+q}}+z,\hfill \\ \hfill N_{m,q}(e_2,z)& ={\displaystyle \frac{2}{{(m+q)}^2}}+{\displaystyle \frac{4}{m+q}}z+{\displaystyle \frac{m+q+1}{m+q}}{z}^2,\hfill \\ \hfill N_{m,q}(e_3,z)& ={\displaystyle \frac{6}{{(m+q)}^3}}+{\displaystyle \frac{18}{{(m+q)}^2}}z+{\displaystyle \frac{9(m+q+1)}{{(m+q)}^2}}{z}^2+{\displaystyle \frac{(m+q+1)(m+q+2)}{{(m+q)}^2}}{z}^3,\hfill \\ \hfill N_{m,q}(e_4,z)& ={\displaystyle \frac{24}{{(m+q)}^4}}+{\displaystyle \frac{96}{{(m+q)}^3}}z+{\displaystyle \frac{72(m+q+1)}{{(m+q)}^3}}{z}^2+{\displaystyle \frac{16(m+q+1)(m+q+2)}{{(m+q)}^3}}{z}^3\hfill \\ \hfill & +{\displaystyle \frac{(m+q+1)(m+q+2)(m+q+3)}{{(m+q)}^3}}{z}^4,\hfill \end{array}$$

Central moments for the operator $N_{m,q}$ can be expressed in terms of the moments of the same operator as follows.

Proposition 2.

For $m=0,1,\dots ,10$, the central moments for the operator $N_{m,q}$ is given by the following:

$$N_{m,q}({(u-z)}^k;z)={\displaystyle \sum _{j=0}^k}{(-z)}^{k-j}\left({\displaystyle \genfrac{}{}{0pt}{}{k}{j}}\right)N_{m,q}(e_k;z),$$

where $N_{m,q}(e_k;z)$ are given in Proposition 1.

Example 2.

For instance, for $k=0,1,\dots ,4$, we have

$$\begin{array}{cc}\hfill N_{m,q}({(u-z)}^0;z)& =1,\hfill \\ \hfill N_{m,q}({(u-z)}^1;z)& ={\displaystyle \frac{1}{m+q}},\hfill \\ \hfill N_{m,q}({(u-z)}^2;z)& ={\displaystyle \frac{2}{{(m+q)}^2}}+{\displaystyle \frac{2}{m+q}}z+{\displaystyle \frac{1}{m+q}}{z}^2,\hfill \\ \hfill N_{m,q}({(u-z)}^3;z)& ={\displaystyle \frac{6}{{(m+q)}^3}}+{\displaystyle \frac{12}{{(m+q)}^2}}z+{\displaystyle \frac{9}{{(m+q)}^2}}{z}^2+{\displaystyle \frac{2}{{(m+q)}^2}}{z}^3,\hfill \\ \hfill N_{m,q}({(u-z)}^4;z)& ={\displaystyle \frac{24}{{(m+q)}^4}}+{\displaystyle \frac{72}{{(m+q)}^3}}z+{\displaystyle \frac{12(m+q+6)}{{(m+q)}^3}}{z}^2\hfill \\ \hfill & +{\displaystyle \frac{4(3m+3q+8)}{{(m+q)}^3}}{z}^3+{\displaystyle \frac{3(m+q+2)}{{(m+q)}^3}}{z}^4.\hfill \end{array}$$

As a consequence of Example 2, we have the following result.

Proposition 3.

For fixed q, we have

1. 

${lim}_{m\to \infty }(m+q)N_{m,q}((u-z);z)=1$,

2. 

${lim}_{m\to \infty }(m+q)N_{m,q}({(u-z)}^2;z)=2z+z^2$,

3. 

${lim}_{m\to \infty }{(m+q)}^2{N}_{m,q}({(u-z)}^3;z)=12z+9z^2+2z^3$,

4. 

${lim}_{m\to \infty }{(m+q)}^2{N}_{m,q}({(u-z)}^4;z)=12z^2+12z^3+3z^4$.

The next result proves the Korovkin-type theorem for the Baskakov–Schurer–Szász operators induced by Sheffer polynomials. The Korovkin-type theorem has been extensively studied in recent years (see, for example, [2,3,15,16,17,18,19,20,21,22,23]).

Theorem 1.

Let $N_{m,q}:C[0,K]\to C[0,K]$ be a sequence of positive linear operators, where $K>0$ is finite. If

$$\underset{m\to \infty }{lim}\parallel N_{m,q}(e_j;z)-e_j\parallel =0,j=0,1,2,$$

where $e_j\left(z\right)=z^j$, then for every $w\in C[0,K]$,

$$\underset{m\to \infty }{lim}\parallel N_{m,q}(w;z)-w\left(z\right)\parallel =0.$$

Proof. 

By Example 1, we obtain $N_{m,q}(e_0;z)=1$, and hence

$$\underset{m\to \infty }{lim}\parallel N_{m,q}(e_0;z)-e_0\parallel =0.$$

Also, $N_{m,q}(e_1;z)={\displaystyle \frac{1}{m+q}}+z$, which implies

$$\underset{m\to \infty }{lim}\parallel N_{m,q}(e_1;z)-e_1\parallel =\underset{m\to \infty }{lim}∥{\displaystyle \frac{1}{m+q}}∥=0.$$

Further, $N_{m,q}(e_2;z)={\displaystyle \frac{2}{{(m+q)}^2}}+{\displaystyle \frac{4}{m+q}}z+{\displaystyle \frac{m+q+1}{m+q}}{z}^2$, therefore

$$\underset{m\to \infty }{lim}\parallel N_{m,q}(e_2;z)-e_2\parallel =0.$$

Hence, by the classical Korovkin theorem on $C[0,K]$ (see [22]), we conclude that

$$\underset{m\to \infty }{lim}\parallel N_{m,q}(w;z)-w\left(z\right)\parallel =0$$

uniformly for every $w\in C[0,K]$. □

3. Direct Estimates

In what follows, we will give an upper bound for the sequence of operators (2).

Theorem 2.

Let $w\in C_B[0,\infty )$. Then the following inequality for the operators (2) holds true: $∥N_{m,q}(w;z)∥\leqq \parallel w\parallel$, for $z\in [0,\infty )$.

Proof. 

From (2) and Example 1, we get

$$\begin{array}{cc}\hfill \left|N_{m,q}(w,z)\right|& \leqq \underset{u\in {\mathbb{R}}^{+}}{sup}\left|w\right(u\left)\right|·(m+q){\displaystyle \sum _{j=0}^{\infty }}{c}_{mq}^j\left(z\right){\int }_0^{\infty }{\displaystyle \frac{b_{mq}^j\left(u\right)}{H^j\left(1\right)·g\left(1\right)}}du\hfill \\ \hfill & =\underset{u\in {\mathbb{R}}^{+}}{sup}\left|w\right(u\left)\right|·N_{m,q}(e_0;z)=\parallel w\parallel ,\hfill \end{array}$$

as asserted by the theorem. □

Let us denote by $B[0,\infty )$, $C[0,\infty )$, and $C_B\left([0,\infty )\right)$, the space of all bounded functions, continuous, and continuous and bounded functions defined in the interval $[0,\infty )$, respectively, endowed with the norm given by $\parallel r\parallel ={sup}_{z\in [0,\infty )}\left|r\right(z\left)\right|$.

The modulus of continuity of the function $r\in C[0,\infty )$ is given as follows:

$$\begin{array}{c}\hfill w(r,\eta ):=sup\left\{\right|r\left(u\right)-r\left(z\right)|:z,u\in [0,\infty \left)\mathrm{and}\right|u-z|\leqq \eta \}.\end{array}$$

It is known that ([24]), for any value of the $|z-u|,$ we have

$$|r\left(z\right)-r\left(u\right)|\leqq \omega (r;\eta )\left({\displaystyle \frac{|z-u|}{\eta }}+1\right).$$

Theorem 3.

Let $w\in C_B[0,\infty )$. Then the following inequality for operators (2) holds true:

$$\begin{array}{c}|N_{m,q}(w;z)-w\left(z\right)|\leqq \omega (w;\eta )\left(1+{\displaystyle \frac{1+2z}{\eta }}\right),\end{array}$$

for $z,\eta \in [0,\infty )$.

Proof. 

Knowing that operators $N_{m,q}$ are linear and positive, then for every $w\in C_B[0,\infty ),$ taking into consideration Example 2, we get

$$\begin{array}{cc}\hfill |N_{m,q}(w;z)-w\left(z\right)|& \leqq (m+q){\displaystyle \sum _{j=0}^{\infty }}{c}_{mq}^j\left(z\right){\int }_0^{\infty }\left|{\displaystyle \frac{b_{mq}^j\left(u\right)}{H^j\left(1\right)·g\left(1\right)}}\right||w\left(u\right)-w\left(z\right)|du\hfill \\ \hfill & \leqq \omega (w;\eta )\left(1+{\displaystyle \frac{1}{\eta }}(m+q){\displaystyle \sum _{j=0}^{\infty }}{c}_{mq}^j\left(z\right){\int }_0^{\infty }\left|{\displaystyle \frac{b_{mq}^j\left(u\right)}{H^j\left(1\right)·g\left(1\right)}}\right|·|u-z|du\right).\hfill \end{array}$$

By the (3) and $u,z>0$, we have

$$\begin{array}{cc}\hfill |N_{m,q}(w;z)-w\left(z\right)|& \leqq \omega (w;\eta )\left(1+{\displaystyle \frac{1}{\eta }}(m+q){\displaystyle \sum _{j=0}^{\infty }}{c}_{mq}^j\left(z\right){\int }_0^{\infty }{\displaystyle \frac{{(m+q)}^j{u}^j{e}^{-(m+q)u}}{j!}}(u+z)du\right)\hfill \\ \hfill & \leqq \omega (w;\eta )\left(1+{\displaystyle \frac{1}{\eta }}{\displaystyle \sum _{j=0}^{\infty }}{c}_{mq}^j\left(z\right)\left({\displaystyle \frac{j+1}{m+q}}+z\right)\right)\hfill \\ \hfill & \leqq \omega (w;\eta )\left(1+{\displaystyle \frac{1}{\eta }}\left({\displaystyle \frac{1}{m+q}}+2z\right)\right).\hfill \end{array}$$

So by the fact that $m\ge 1$ and $q>0$, we have

$$\begin{array}{cc}\hfill |N_{m,q}(w;z)-w\left(z\right)|& \leqq \omega (w;\eta )\left(1+{\displaystyle \frac{1+2z}{\eta }}\right).\hfill \end{array}$$

which completes the proof. □

For $w\in C[0,\infty )$ and $\eta >0$, the second-order modulus of smoothness of r is defined as follows:

$$\begin{array}{c}\hfill omeg{a}_2(w,\sqrt{\eta })=\underset{0<l\leqq \sqrt{\eta }}{sup}\underset{z,z+l\in [0,\infty )}{sup}\left\{\right|w(z+2l)-2w\left(z\right)+w(z-l)\left|\right\}.\end{array}$$

The Peetre’s K-functional [22] is defined by $K_2(w,\eta )=inf\left\{{\parallel w-t\parallel }_{C[0,\infty )}+\eta {\parallel t^{″}\parallel }_{C[0,\infty )}:t\in W^2\right\}$, where $\eta >0$ and $W^2=\{t\in C[0,\infty ):t^{\prime },t^{″}\in C[0,\infty )\}$. It is known that there exists a positive constant $\mathcal{C}>0$ such that $K_2(w,\eta )\leqq \mathcal{C}{\omega }_2(r,\sqrt{\eta })$ with $\eta >0$ (see [25], Theorem 3.1.2).

Theorem 4.

Let $w\in C[0,K]$ for any finite real number K. Then

$$\left|N_{m,q}(w;z)-w\left(z\right)\right|\leqq {\displaystyle \frac{2}{A}}\parallel w\parallel c^2+{\displaystyle \frac{3}{4}}(A+c^2+2){\omega }_2(w;c),$$

where $c=\sqrt[4]{N_{m,q}\left({(u-z)}^2;z\right)}$.

The proof follows along the same lines as the proof of Theorem 5 in [26], and is therefore omitted.

The Voronovskaya-type theorem for the Baskakov–Schurer–Szász operators induced by Sheffer polynomials is given by the following result:

Theorem 5.

For $w,w^{\prime },w^{″}\in C_B[0,\infty ),$ the following limit relation:

$$\underset{m\to \infty }{lim}m\left[N_{m,q}(w;z)-w\left(z\right)\right]=w^{\prime }\left(z\right)+{\displaystyle \frac{w^{″}\left(z\right)}{2}}(z+z^2),$$

holds true for every $z\in [0,K]$ and any finite K.

Proof. 

By Taylor’s expansion of the function $w\in C_B[0,\infty )$, we have

$$w\left(u\right)=w\left(z\right)+(u-z)w^{\prime }\left(z\right)+{\displaystyle \frac{1}{2}}{(u-z)}^2{w}^{″}\left(z\right)+{(u-z)}^2{\kappa }_z\left(u\right),$$

where

$${\kappa }_z\left(u\right)=\left\{\begin{array}{cc}{\displaystyle \frac{w\left(u\right)-w\left(z\right)-(u-z)w^{\prime }\left(z\right)-\frac{1}{2}{(u-z)}^2{w}^{″}\left(z\right)}{{(u-z)}^2}}\hfill & (z\ne u)\hfill \\ 0\hfill & (z=u)\hfill \end{array}\right.$$

and the function ${\kappa }_z(·)$ is the Peano form of the remainder, ${\kappa }_z(·)\in C_B[0,\infty )$ and ${\kappa }_z\left(u\right)\to 0$ as $u\to z.$ Applying the operator $N_{m,q}$ on both sides of the above relation, it yields

$$\begin{array}{cc}\hfill & m[N_{m,q}(w;z)-w\left(z\right)]\hfill \\ \hfill & =w^{\prime }\left(z\right)m{N}_{m,q}((u-z);z)+{\displaystyle \frac{w^{″}\left(z\right)}{2}}m{N}_{m,q}({(u-z)}^2;z)+m{N}_{m,q}({(u-z)}^2{\kappa }_z\left(u\right);z).\hfill \end{array}$$

Taking into consideration Example 2, we obtain

$$\begin{array}{cc}\hfill & \underset{m\to \infty }{lim}m\left[N_{m,q}(w;z)-w\left(z\right)\right]\hfill \\ \hfill & =w^{\prime }\left(z\right)+{\displaystyle \frac{w^{″}\left(z\right)}{2}}\left(z+z^2\right)+\underset{m\to \infty }{lim}m{N}_{m,q}({(u-z)}^2{\kappa }_z\left(u\right);z),\hfill \end{array}$$

and applying the Cauchy-Schwarz inequality, yields

$$m{N}_{m,q}({(u-z)}^2{\kappa }_z\left(u\right);z)\leqq {\left\{m^2{N}_{m,q}({(u-z)}^4;z){\}}^{{\displaystyle \frac{1}{2}}}\{N_{m,q}({\kappa }_z^2\left(u\right);z)\right\}}^{{\displaystyle \frac{1}{2}}}.$$

We now observe that ${\kappa }_z^2\left(u\right)\to 0$ as $u\to z$ and ${\kappa }_z^2(·)\in C_B[0,\infty ).$ So, from Example 2, it follows that

$$m{N}_{m,q}({(u-z)}^2{\kappa }_z\left(u\right);z)\to 0,$$

as $m\to \infty ,$ for every $z\in [0,K].$ From last relations we get that

$$\begin{array}{c}\hfill \underset{m\to \infty }{lim}m\left[N_{m,q}(w;z)-w\left(z\right)\right]=w^{\prime }\left(z\right)+{\displaystyle \frac{w^{″}\left(z\right)}{2}}(z+z^2).\end{array}$$

This completes the proof. □

Theorem 5 describes the asymptotic behavior of the approximation error $N_{m,q}(w;z)-w\left(z\right)$, and therefore it represents a Voronovskaya-type theorem for the operators $N_{m,q}$. We next investigate the asymptotic behavior of the quantity $N_{m,q}(rt;z)-N_{m,q}(r;z)N_{m,q}(t;z)$, which measures the deviation of the operators from the multiplicative property. Results of this type are called Grüss–Voronovskaya-type theorems, since they combine the asymptotic nature of Voronovskaya-type results with the Grüss-type analysis of product deviations. In what follows, we will give the Grüss–Voronovskaya-type theorem (see [27]) for the Baskakov–Schurer–Szász operators induced by Sheffer polynomials.

Theorem 6.

Let $r,r^{\prime },r^{″},t,t^{\prime },t^{″}\in C_B[0,\infty ).$ Then

$$\underset{m\to \infty }{lim}m\left|N_{m,q}(rt;z)-N_{m,q}(r,z)N_{m,q}(t;z)\right|=(z+z^2)r^{\prime }\left(z\right)t^{\prime }\left(z\right),$$

for each $z\in [0,K],$ where K is finite.

Proof. 

After some calculations, we obtain

$$\begin{array}{cc}\hfill & m\left(N_{m,q}(rt,z)-N_{m,q}(r,z)N_{m,q}(t,z)\right)\hfill \\ \hfill & =\left[m\left(N_{m,q}(rt;z)-rt\right)-{\left(rt\right)}^{\prime }\left(z\right)-(z+z^2){\displaystyle \frac{{\left(rt\right)}^{″}\left(z\right)}{2}}\right]\hfill \\ \hfill & -t\left(z\right)\left[m\left(N_{m,q}(r,z)-r\left(z\right)\right)-r^{\prime }\left(z\right)-(z+z^2){\displaystyle \frac{r^{″}\left(z\right)}{2}}\right]\hfill \\ \hfill & -N_{m,q}(r;z)\left[m\left(N_{m,q}(t,z)-t\left(z\right)\right)-t^{\prime }\left(z\right)-(z+z^2){\displaystyle \frac{t^{″}\left(z\right)}{2}}\right]\hfill \\ \hfill & +\left(z+z^2\right)r^{\prime }\left(z\right)t^{\prime }\left(z\right)+\left(z+z^2\right){\displaystyle \frac{t^{″}\left(z\right)}{2}}[r\left(z\right)-N_{m,q}(r;z)]\hfill \\ \hfill & +t^{\prime }\left(z\right)[r\left(z\right)-N_{m,q}(r;z)].\hfill \end{array}$$

Therefore, the proof of the theorem follows from Theorems 1 and 5. □

The following result expresses the Grüss–Voronovskaya-type theorem, for the fractional form of the functions, for the Baskakov–Schurer–Szász operators induced by Sheffer polynomials.

Theorem 7.

Let $r^{\prime },r^{″},t^{\prime },t^{″}\in C_B[0,\infty ).$ Then

$$\begin{array}{cc}\hfill & \underset{m\to \infty }{lim}m\left|N_{m,q}\left({\displaystyle \frac{r}{t}};z\right)-{\displaystyle \frac{N_{m,q}(r,z)}{N_{m,q}(t;z)}}\right|=(z+z^2)\left(-{\displaystyle \frac{r^{\prime }{t}^{\prime }}{t^2}}+{\displaystyle \frac{r{t}^{\prime 2}}{t^3}}\right)+{\displaystyle \frac{r\left(z\right)t^{\prime }\left(z\right)}{t^2\left(z\right)}},\hfill \end{array}$$

for each $z\in [0,K],$ where K is finite and $t\left(z\right)\ne 0.$

Proof. 

After some calculations, we obtain

$$\begin{array}{cc}\hfill & m\left(N_{m,q}\left({\displaystyle \frac{r}{t}},z\right)-{\displaystyle \frac{N_{m,q}(r,z)}{N_{m,q}(t,z)}}\right)\hfill \\ \hfill & =\left[m\left(N_{m,q}\left({\displaystyle \frac{r}{t}},z\right)-{\displaystyle \frac{r}{t}}\left(z\right)\right)-{\left({\displaystyle \frac{r}{t}}\left(z\right)\right)}^{\prime }-(z+z^2){\displaystyle \frac{{\left(\frac{r}{t}\left(z\right)\right)}^{″}}{2}}\right]\hfill \\ \hfill & -{\displaystyle \frac{1}{t\left(z\right)}}\left(m\left[N_{m,q}(r,z)-r\left(z\right)\right]-r^{\prime }\left(z\right)-(z+z^2){\displaystyle \frac{r^{″}\left(z\right)}{2}}\right)\hfill \\ \hfill & +{\displaystyle \frac{N_{m,q}(r;z)}{t\left(z\right)N_{m,q}(t,z)}}\left[m(N_{m,q}(t,z)-t\left(z\right))-t^{\prime }\left(z\right)-(z+z^2){\displaystyle \frac{t^{″}\left(z\right)}{2}}\right]\hfill \\ \hfill & +(z+z^2)\left(-{\displaystyle \frac{r{t}^{″}}{2t^2}}-{\displaystyle \frac{r^{\prime }{t}^{\prime }}{t^2}}+{\displaystyle \frac{r{t}^{\prime 2}}{t^3}}\right)+{\displaystyle \frac{N_{m,q}(r,z)}{N_{m,q}(t,z)}}\left[{\displaystyle \frac{t^{\prime }\left(z\right)}{t\left(z\right)}}+(z+z^2){\displaystyle \frac{t^{″}\left(z\right)}{2t\left(z\right)}}\right].\hfill \end{array}$$

(4)

From Proposition 3, we know that

$$N_{m,q}({(u-z)}^4,z)\sim {\displaystyle \frac{12z^2+12z^3+3z^4}{{(m+q)}^2}},$$

as $m\to \infty .$ On the other hand, by Taylor’s expansion theorem for the functions $r,t\in C_B[0,\infty ),$ we obtain the following:

$$r\left(u\right)=r\left(z\right)+(u-z)r^{\prime }\left(z\right)+{\displaystyle \frac{1}{2}}{(u-z)}^2{r}^{″}\left(z\right)+{(u-z)}^2{\kappa }_z\left(u\right),$$

and

$$t\left(u\right)=t\left(z\right)+(u-z)t^{\prime }\left(z\right)+{\displaystyle \frac{1}{2}}{(u-z)}^2{t}^{″}\left(z\right)+{(u-z)}^2{\nu }_z\left(u\right),$$

where

$${\kappa }_z\left(u\right)=\left\{\begin{array}{cc}{\displaystyle \frac{r\left(u\right)-r\left(z\right)-(u-z)r^{\prime }\left(z\right)-\frac{1}{2}{(u-z)}^2{r}^{″}\left(z\right)}{{(u-z)}^2}}\hfill & (z\ne u)\hfill \\ 0\hfill & (z=u)\hfill \end{array}\right.$$

the function ${\kappa }_z(·)$ is the Peano form of the remainder, ${\kappa }_z(·)\in C_B[0,\infty ),$${\kappa }_z\left(u\right)\to 0$ as $u\to z.$ And

$${\nu }_z\left(u\right)=\left\{\begin{array}{cc}{\displaystyle \frac{t\left(u\right)-t\left(z\right)-(u-z)t^{\prime }\left(z\right)-\frac{1}{2}{(u-z)}^2{t}^{″}\left(z\right)}{{(u-z)}^2}}\hfill & (z\ne u)\hfill \\ 0\hfill & (z=u)\hfill \end{array}\right.$$

the function ${\nu }_z(·)$ is the Peano form of the remainder, ${\nu }_z(·)\in C_B[0,\infty ),$${\nu }_z\left(u\right)\to 0$ as $u\to z.$ From the last relations, we obtain the following:

$$\begin{array}{cc}\hfill \underset{m\to \infty }{lim}{\displaystyle \frac{N_{m,q}(r,z)}{N_{m,q}(t,z)}}& =\underset{m\to \infty }{lim}{\displaystyle \frac{r\left(z\right)+r^{\prime }\left(z\right)\frac{1}{m+q}+\frac{r^{″}\left(z\right)}{2}\frac{2z+z^2}{m+q}+N_{m,q}({(u-z)}^2{\kappa }_z\left(u\right);z)}{t\left(z\right)+t^{\prime }\left(z\right)\frac{1}{m+q}+\frac{t^{″}\left(z\right)}{2}\frac{2z+z^2}{m+q}+N_{m,q}({(u-z)}^2{\nu }_z\left(u\right);z)}}\hfill \\ \hfill & =\underset{m\to \infty }{lim}{\displaystyle \frac{r\left(z\right)+N_{m,q}({(u-z)}^2{\kappa }_z\left(u\right);z)}{t\left(z\right)+N_{m,q}({(u-z)}^2{\nu }_z\left(u\right);z)}}.\hfill \end{array}$$

After applying Cauchy–Schwarz inequality, it yields

$$N_{m,q}({(u-z)}^2{\kappa }_z\left(u\right);z)\leqq {\left\{N_{m,q}({(u-z)}^4;z){\}}^{{\displaystyle \frac{1}{2}}}\{N_{m,q}({\kappa }_z^2\left(u\right);z)\right\}}^{{\displaystyle \frac{1}{2}}}.$$

Then it follows that ${lim}_{m\to \infty }{N}_{m,q}({(u-z)}^2{\kappa }_z\left(u\right);z)=0$, ${lim}_{m\to \infty }{N}_{m,q}({(u-z)}^2{\nu }_z\left(u\right);z)=0$, and

$$\underset{m\to \infty }{lim}{\displaystyle \frac{N_{m,q}(r,z)}{N_{m,q}(t,z)}}={\displaystyle \frac{r\left(z\right)}{t\left(z\right)}}.$$

Based on the above relations, Theorem 1, Theorem 6 and passing by limit where $m\to \infty ,$ in relation (4), we get

$$\begin{array}{cc}\hfill & \underset{m\to \infty }{lim}m\left(N_{m,q}\left({\displaystyle \frac{r}{t}},z\right)-{\displaystyle \frac{N_{m,q}(r,z)}{N_{m,q}(t,z)}}\right)\hfill \\ \hfill & =(z+z^2)\left(-{\displaystyle \frac{r^{\prime }{t}^{\prime }}{t^2}}+{\displaystyle \frac{r{t}^{\prime 2}}{t^3}}\right)+{\displaystyle \frac{r\left(z\right)t^{\prime }\left(z\right)}{t^2\left(z\right)}},\hfill \end{array}$$

as required. □

The following result gives the rate of the speed of the change between the difference of the Baskakov–Schurer–Szász operators induced by Sheffer polynomials, with its function and their derivatives, measured by the modulus of continuity.

Theorem 8.

Let $w,w^{\prime },w^{″}\in C[0,\infty ).$ Then

$$\begin{array}{cc}\hfill & |(m+q)\left[N_{m,q}(w;z)-w\left(z\right)\right]-w^{\prime }\left(z\right)-{\displaystyle \frac{w^{″}\left(z\right)}{2}}\left[{\displaystyle \frac{2}{(m+q)}}+2z+z^2\right]|=O\left(1\right)\omega \left(w^{″};{\displaystyle \frac{1}{\sqrt{m}}}\right),\hfill \end{array}$$

as $m\to \infty$, for every $z\in [0,K],$ and for any finite $K.$

Proof. 

From Taylor’s theorem, we have

$$w\left(u\right)=w\left(z\right)+w^{\prime }\left(z\right)(u-z)+{\displaystyle \frac{w^{″}\left(z\right)}{2}}{(u-z)}^2+R(u,z),$$

where $R(u,z)={\displaystyle \frac{w^{″}\left(\eta \right)-w^{″}\left(z\right)}{2}}{(u-z)}^2$, for $\eta \in (u,z).$ We thus find that

$$\begin{array}{cc}\hfill & \left|N_{m,q}(w;z)-w\left(z\right)-w^{\prime }\left(z\right)N_{m,q}(u-z;z)-{\displaystyle \frac{w^{″}\left(z\right)}{2}}{N}_{m,q}({(u-z)}^2;z)\right|\hfill \\ \hfill & \leqq N_{m,q}\left(\right|R(u,z)|;z).\hfill \end{array}$$

So,

$$\begin{array}{cc}\hfill & |(m+q)\left[N_{m,q}(w;z)-w\left(z\right)\right]-w^{\prime }\left(z\right)-{\displaystyle \frac{w^{″}\left(z\right)}{2}}\left[{\displaystyle \frac{2}{(m+q)}}+2z+z^2\right]|\hfill \\ \hfill & \leqq (m+q)N_{m,q}\left(\left|R\right(u,z\left)\right|;z\right).\hfill \end{array}$$

From the properties of the modulus of continuity, it yields

$$\left|{\displaystyle \frac{w^{″}\left(\eta \right)-w^{″}\left(z\right)}{2!}}\right|\leqq {\displaystyle \frac{1}{2!}}\left(1+{\displaystyle \frac{|\eta -z|}{\gamma }}\right)\omega (w^{″};\gamma ),$$

for some $\gamma >0.$ On the other hand, we get

$$\left|{\displaystyle \frac{w^{″}\left(\eta \right)-w^{″}\left(z\right)}{2!}}\right|\leqq \left\{\begin{array}{cc}\omega (w^{″};\gamma )\hfill & \left(\right|\eta -z|\leqq \gamma )\hfill \\ {\displaystyle \frac{{(\eta -z)}^2}{{\gamma }^2}}\omega (w^{″};\gamma )\hfill & \left(\right|\eta -z|\geqq \gamma ).\hfill \end{array}\right.$$

For $0<\gamma <1,$ we obtain that

$$\left|{\displaystyle \frac{w^{″}\left(\eta \right)-w^{″}\left(z\right)}{2!}}\right|\leqq \omega (w^{″};\gamma )\left(1+{\displaystyle \frac{{(\eta -z)}^2}{{\gamma }^2}}\right),$$

and

$$\left|R(u,z)\right|\leqq \omega (w^{″};\gamma )\left(1+{\displaystyle \frac{{(u-z)}^2}{{\gamma }^2}}\right){(u-z)}^2=\omega (w^{″};\gamma )\left({(u-z)}^2+{\displaystyle \frac{{(u-z)}^4}{{\gamma }^4}}\right).$$

By the linearity of $N_{m,q}$ and the above relation, we get

$$N_{m,q}\left(\right|R(u,z)|;z)\leqq \omega (w^{″};\gamma )\left(N_{m,q}({(u-z)}^2;z)+{\displaystyle \frac{1}{{\gamma }^4}}{N}_{m,q}({(u-z)}^4;z)\right).$$

Now, from Example 2, it follows that

$$N_{m,q}\left(\right|R(u,z)|;z)\leqq \omega (w^{″};\gamma )\left(O\left({\displaystyle \frac{1}{m}}\right)+{\displaystyle \frac{1}{{\gamma }^4}}O\left({\displaystyle \frac{1}{m^3}}\right)\right)=O\left({\displaystyle \frac{1}{m}}\right)\omega (w^{″};\gamma ).$$

Thus, for $\gamma ={\displaystyle \frac{1}{\sqrt{m}}}$, we complete the proof. □

The Ditzian–Totik uniform modulus of smoothness of the first and the second orders are defined as follows (see [25]):

$$\begin{array}{c}\hfill {\omega }_{\gamma }(r;\eta ):=\underset{0<\left|h\right|\leqq \eta }{sup}\underset{z,z+h\gamma \left(z\right)\in [0,\infty )}{sup}\left\{|r\left(z+h\gamma \left(z\right)\right)-r\left(z\right)|\right\}\end{array}$$

and

$$\begin{array}{c}\hfill {\omega }_2^{\psi }(r;\eta ):=\underset{0<\left|h\right|\leqq \eta }{sup}\underset{z,z\pm h\psi \left(z\right)\in [0,\infty )}{sup}\left\{|r\left(z+h\psi \left(z\right)\right)-2r\left(z\right)+r\left(z-h\psi \left(z\right)\right)|\right\},\end{array}$$

respectively, where $\psi$ is an admissible step-weight function on $[c,d]$, that is,

$$\psi \left(z\right)={\left[(z-c)(d-z)\right]}^{1/2},$$

with $z\in [c,d].$

The corresponding K-functional is defined as follows:

$$\begin{array}{c}\hfill K_{2,\psi \left(z\right)}(r,\eta )=\underset{t\in W^2\left(\psi \right)}{inf}\left\{{\parallel r-t\parallel }_{C[0,\infty )}+\eta {\parallel {\psi }^2{t}^{″}\parallel }_{C[0,\infty )}\right\},\end{array}$$

where $\eta >0$ and

$$W^2\left(\psi \right)=\{t\in C_B[0,\infty ):t^{\prime }\in AC[0,\infty ),{\psi }^2{t}^{″}\in C_B[0,\infty )\}\mathrm{and}{t}^{\prime }\in AC[0,\infty )$$

means that $t^{\prime }$ is absolutely continuous on $[0,\infty )$.

Theorem 9.

Let $\psi =\sqrt{z(1-z)}(z\in [0,1])$ be a step-weight function then, for any $w\in C_B[0,\infty )$ and $z\in [0,1],$

$$\begin{array}{cc}\hfill \parallel N_{m,q}(w;z)-w\left(z\right)\parallel & \leqq K_{2,\psi \left(z\right)}\left(w,{\displaystyle \frac{N_{m,q}\left({(u-z)}^2;z\right)+1}{4{\psi }^2\left(z\right)}}\right)+{\omega }_{\gamma }\left(w;{\displaystyle \frac{1}{\gamma \left(z\right)}}\right).\hfill \end{array}$$

Proof. 

Let $N_{m,q}^{*}(w;z)=N_{m,q}(w;z)+w\left(z\right)-w\left(z+{\gamma }_1(m,q)\right)$, where ${\gamma }_1(m,q)={\displaystyle \frac{1}{m+q}}$. Note that $N_{m,q}^{*}(1;z)=1$ and $N_{m,q}^{*}\left((u-z);z\right)=0$. Let $t\in W^2\left(\psi \right)$, Then, by using Taylor’s expansion, we write

$$t\left(u\right)=t\left(z\right)+t^{\prime }\left(z\right)(u-z)+{\int }_z^u(u-v)t^{″}\left(v\right)\mathrm{d}v,u\in [0,\infty ),$$

which implies that

$$\begin{array}{cc}\hfill & N_{m,q}^{*}(t;z)-t\left(z\right)\hfill \\ \hfill & =N_{m,q}\left({\int }_z^u(u-v)t^{″}\left(v\right)\mathrm{d}v;z\right)-{\int }_z^{z+{\gamma }_1(m,q)}[z+{\gamma }_1(m,q)-v]t^{″}\left(v\right)\mathrm{d}v.\hfill \end{array}$$

Based on the above relations, we get

$$\begin{array}{cc}\hfill & |N_{m,q}^{*}(t;z)-t\left(z\right)|\hfill \\ \hfill & \leqq N_{m,q}\left(\left|{\int }_z^u(u-v)t^{″}\left(v\right)\mathrm{d}v\right|;z\right)+{\int }_z^{z+{\gamma }_1(m,q)}|z+{\gamma }_1(m,q)-v|·|t^{″}\left(v\right)|\mathrm{d}v\hfill \\ \hfill & \leqq ∥{\psi }^2{t}^{″}\left(z\right)N_{m,q}\left(\left|{\int }_z^u{\displaystyle \frac{|u-v|}{{\psi }^2\left(v\right)}}\mathrm{d}v\right|;z\right)+\parallel {\psi }^2{t}^{″}\left(z\right)\parallel ∥\left|{\int }_z^{z+{\gamma }_1(m,q)}{\displaystyle \frac{|z+{\gamma }_1(m,q)-v|}{{\psi }^2\left(v\right)}}\mathrm{d}v\right|.\hfill \end{array}$$

Let $v=\mu z+(1-\mu )u$$(\mu \in [0,1\left]\right)$. Since ${\psi }^2$ is concave on $[0,\infty )$, it follows that ${\psi }^2\left(v\right)\ge \mu {\psi }^2\left(z\right)+(1-\mu ){\psi }^2\left(u\right)$ and hence

$$\begin{array}{c}\hfill {\displaystyle \frac{|u-v|}{{\psi }^2\left(v\right)}}={\displaystyle \frac{\mu |z-u|}{{\psi }^2\left(v\right)}}\leqq {\displaystyle \frac{\mu |z-u|}{\mu {\psi }^2\left(z\right)+(1-\mu ){\psi }^2\left(u\right)}}\leqq {\displaystyle \frac{|z-u|}{{\psi }^2\left(z\right)}}.\end{array}$$

Then, it yields

$$\parallel N_{m,q}^{*}\left(t\right)-t\parallel \leqq {\displaystyle \frac{\parallel {\psi }^2{t}^{″}{\parallel }_{C[0,\infty )}}{{\psi }^2\left(z\right)}}\left\{\left[N_{m,q}\left({(u-z)}^2;z\right)\right]+z{\gamma }_1(m,q)\right\}.$$

From the above relations, we obtain

$$\begin{array}{cc}\hfill & \parallel N_{m,q}^{*}(w,z)-w\left(z\right)\parallel \hfill \\ \hfill & \leqq \parallel N_{m,q}^{*}(w-t)\parallel +\parallel N_{m,q}^{*}\left(t\right)-t\parallel +\parallel w-t\parallel +\parallel w\left(z+{\gamma }_1(m,q)\right)-w\left(z\right)\parallel \hfill \\ \hfill & \leqq 4\parallel w-t\parallel +{\displaystyle \frac{\parallel {\psi }^2{t}^{″}\parallel }{{\psi }^2\left(z\right)}}\left[N_{m,q}\left({(u-z)}^2;z\right)+z{\gamma }_1(m,q)\right]+\parallel w(z+{\gamma }_1(m,q)-w\left(t\right)\parallel .\hfill \end{array}$$

We know that

$$\begin{array}{cc}\hfill \parallel w(z+{\gamma }_1(m,q))-w\left(z\right)\parallel & \leqq ∥w\left(z+\gamma \left(z\right){\displaystyle \frac{N_{m,q}\left((u-z);z\right)}{\gamma \left(z\right)}}\right)-w\left(z\right)∥\hfill \\ \hfill & \leqq {\omega }_{\gamma }\left(w;{\displaystyle \frac{{\gamma }_1(m,q)}{\gamma \left(z\right)}}\right).\hfill \end{array}$$

Therefore, we have

$$\begin{array}{cc}\hfill \parallel N_{m,q}(w,z)-w\left(z\right)\parallel & \leqq 4K_{2,\psi \left(z\right)}\left(w,{\displaystyle \frac{N_{m,q}\left({(u-z)}^2;z\right)+z{\gamma }_1(m,q)}{4{\psi }^2\left(z\right)}}\right)+{\omega }_{\gamma }\left(w;{\displaystyle \frac{{\gamma }_1(m,q)}{\gamma \left(z\right)}}\right).\hfill \end{array}$$

From the conditions given in the theorem, the properties of the K-functional and the modulus of continuity, we get

1.

${\displaystyle \frac{1}{m+q}}\le 1,$

2.

$K_{2,\psi \left(z\right)}\left(w,{\displaystyle \frac{N_{m,q}\left({(u-z)}^2;z\right)+z{\gamma }_1(m,q)}{4{\psi }^2\left(z\right)}}\right)\leqq K_{2,\psi \left(z\right)}\left(w,{\displaystyle \frac{N_{m,q}\left({(u-z)}^2;z\right)+1}{4{\psi }^2\left(z\right)}}\right),$

3.

${\omega }_{\gamma }\left(w;{\displaystyle \frac{{\gamma }_1(m,q)}{\gamma \left(z\right)}}\right)\leqq {\omega }_{\gamma }\left(w;{\displaystyle \frac{1}{\gamma \left(z\right)}}\right)$

for every $z\in [0,1].$ From the last relations, it yields the following:

$$\begin{array}{cc}\hfill \parallel N_{m,q}(w;z)-w\left(z\right)\parallel & \leqq K_{2,\psi \left(z\right)}\left(w,{\displaystyle \frac{N_{m,q}\left({(u-z)}^2;z\right)+1}{4{\psi }^2\left(z\right)}}\right)+{\omega }_{\gamma }\left(w;{\displaystyle \frac{1}{\gamma \left(z\right)}}\right),\hfill \end{array}$$

as asserted by the theorem. □

4. Weighted Approximation and Shape-Preserving Properties

Let $\tau \left(z\right)=z^2+1$ and $B_r$ be a positive constant. We define the weighted space of functions as follows:

(i)

$B_{\tau }[0,\infty )$ is the space of functions r defined on $[0,\infty )$ with property $\left|r\left(z\right)\right|\leqq B_r\tau \left(z\right).$

(ii)

$C_{\tau }[0,\infty )\subset B_{\tau }[0,\infty ).$

(iii)

$C_{\tau }^{*}[0,\infty )\subset C_{\tau }[0,\infty )$ and for $r\in C_{\tau }[0,\infty ),$ then ${lim}_{z\to \infty }{\displaystyle \frac{r\left(z\right)}{\tau \left(z\right)}}<\infty .$

The $B_{\tau }[0,\infty ),$ is normed space with norm given by

$${\parallel r\parallel }_{\tau }=\underset{z\geqq 0}{sup}{\displaystyle \frac{\left|r\right(z\left)\right|}{\tau \left(z\right)}}.$$

In the sequel, we give the following weighted modulus of continuity $\Omega (r;\eta )$ defined on the infinite interval $[0,\infty )$ as

$$\Omega (r;\eta )=\underset{z\geqq 0;0<\left|l\right|\leqq \eta }{sup}{\displaystyle \frac{\left|r\right(z+l)-r(z\left)\right|}{(1+l^2)\tau \left(z\right)}}\left(\forall r\in C_{\tau }^{*}[0,\infty )\right).$$

For any $j\in [0,\infty )$, the weighted modulus of continuity $\Omega (r;\eta )$ satisfies the following inequality:

$$\begin{array}{c}\hfill \Omega (r;j\eta )\leqq 2(1+j)(1+{\eta }^2)\Omega (r;\eta ),\end{array}$$

and, for every $r\in C_{\tau }^{*}[0,\infty ),$ we get

$$\begin{array}{c}\hfill |r\left(u\right)-r\left(z\right)|\leqq 2\left({\displaystyle \frac{|u-z|}{\eta }}+1\right)(1+{\eta }^2)\Omega (r;\eta )(1+z^2)(1+{(u-z)}^2).\end{array}$$

Theorem 10.

Let $w\in C_{\tau }^{*}[0,\infty ),$ then

$$\begin{array}{c}\hfill \underset{m\to \infty }{lim}\parallel N_{m,q}(w;z)-w\left(z\right){\parallel }_{\tau }=0.\end{array}$$

Proof. 

It is enough to check that $N_{m,q}(e_j;z)$ converges uniformly to $e_j,$ for $j\in \{0,1,2\}$ as $m\to \infty$ and then we can apply the well-known weighted Korovkin type theorem, where $e_j\left(z\right)=z^j$. So we will prove that

$$\underset{m}{lim}\parallel N_{m,q}(e_j;z)-e_j\left(z\right){\parallel }_{\tau }=0,j=0,1,2.$$

The case where $j=0,$ follows immediately from Example 1. In what follows, we will prove the above relation for $j=1$ and $j=2$, respectively. For $w\in C_{\tau }^{*}[0,\infty )$ and Example 1, we get

$$\begin{array}{c}\hfill \parallel N_{m,q}(e_1;z)-e_1\left(z\right){\parallel }_{\tau }=\underset{z\geqq 0}{sup}\left\{{\displaystyle \frac{|N_{m,q}(e_1;z)-e_1\left(z\right)|}{\tau \left(z\right)}}\right\}\leqq \underset{z\geqq 0}{sup}{\displaystyle \frac{1}{(m+q)\tau \left(z\right)}}=0\end{array}$$

and

$$\begin{array}{c}\hfill \parallel N_{m,q}(e_2;z)-e_2\left(z\right){\parallel }_{\tau }=\underset{z\geqq 0}{sup}\left\{{\displaystyle \frac{|N_{m,q}(e_2;z)-e_2\left(z\right)|}{\tau \left(z\right)}}\right\}\leqq \underset{z\geqq 0}{sup}\left\{{\displaystyle \frac{\left|\frac{2}{{(m+q)}^2}+\frac{4z}{m+q}+\frac{z^2}{m+q}\right|}{\tau \left(z\right)}}\right\}=0.\end{array}$$

From the above relations and the Korovkin-type theorem, we obtain

$$\underset{m\to \infty }{lim}\parallel N_{m,q}(e_j;z)-e_j\left(z\right){\parallel }_{\tau }=0,j=0,1,2,$$

which completes the proof. □

In the sequel, we will prove that operators defined by relation (1) are shape-preserving.

Theorem 11.

Let $w\in C^2[0,\infty )$. If $w\left(z\right)$ is convex on $[0,\infty ).$ Then the operators defined by relation (2), are also convex.

Proof. 

Let us suppose that $w\left(z\right)$ is convex and that $z_0$ and $z_1$ are distinct points in the interval $[a,b],$ where $a<z_0<z_1<b$ and $a,b\in [c,d]\subset [0,\infty ).$ Then the Lagrangian interpolation polynomial is: $\left(z_0,w\left(z_0\right)\right)$ and $\left(z_1,w\left(z_1\right)\right)$ is given by

$$Q\left(z\right)={\displaystyle \frac{z-z_1}{z_0-z_1}}w\left(z_0\right)+{\displaystyle \frac{z-z_0}{z_1-z_0}}w\left(z_1\right).$$

Then, from Example 1, it yields

$$\begin{array}{cc}\hfill {\left[N_{m,q}(Q;z)\right]}^{″}& ={\left[{\displaystyle \frac{w\left(z_0\right)}{z_0-z_1}}+{\displaystyle \frac{w\left(z_1\right)}{z_1-z_0}}\right]}^{″}=0.\hfill \end{array}$$

On the other side, it follows that

$$\begin{array}{cc}\hfill N_{m,q}(w;z)& =N_{m,q}(Q;z)\hfill \\ \hfill & +{\displaystyle \frac{w^{″}\left({\mu }_z\right)}{2!}}\left[N_{m,q}(u^2;z)-(z_0+z_1)N_{m,q}(u,z)+z_0{z}_1\right]\hfill \\ \hfill & =N_{m,q}(Q;z)\hfill \\ \hfill & +{\displaystyle \frac{w^{″}\left({\mu }_z\right)}{2!}}\left[{\displaystyle \frac{2}{{(m+q)}^2}}+{\displaystyle \frac{4}{m+q}}z+{\displaystyle \frac{m+q+1}{m+q}}{z}^2-(z_0+z_1)\left({\displaystyle \frac{1}{m+q}}+z\right)+z_0{z}_1\right].\hfill \end{array}$$

From the above relations and given conditions, we obtain

$${\left[N_{m,q}(w;z)\right]}^{″}=2w^{″}\left({\mu }_z\right)·{\displaystyle \frac{m+q+1}{m+q}}>0,$$

which proves our theorem. □

5. Concluding Remarks

In this paper, we introduced a new class of Baskakov–Schurer–Szász operators generated by Sheffer polynomials in the particular case where $s_k\left(x\right)=g\left(1\right){\left(H\left(1\right)\right)}^k{x}^k$. We established a Korovkin-type theorem. Furthermore, we derived error estimates in terms of the Ditzian–Totik modulus of smoothness, as well as weighted approximation results and shape-preserving properties.

It is interesting to study Baskakov–Schurer–Szász operators generated by Sheffer polynomials in the general case and to establish a Korovkin-type theorem, as well as to derive error estimates in terms of the Ditzian–Totik modulus of smoothness.

Another possible direction for future work is to investigate similar approximation and shape-preserving properties for other generalized operators or to study applications of the present operators in numerical analysis and related areas.