A Dunkl–Type Generalization of Szász–Kantorovich Operators via Post–Quantum Calculus

Abstract

In this paper, we define the $(p,q)$-variant of Szász–Kantorovich operators via Dunkl-type generalization generated by an exponential function and study the Korovkin-type results. We also obtain the convergence of our operators in weighted space by the modulus of continuity, Lipschitz class, and Peetre’s K-functionals. The extra parameter p provides more flexibility in approximation and plays an important role in symmetrizing these newly-defined operators.

1. Introduction and Preliminaries

Bernstein [1] and q-Bernstein ([2,3]) operators have become very important tools in the study of approximation theory and several branches of applied sciences and engineering. For ${\left[r\right]}_{p,q}=\frac{p^r-q^r}{p-q},r=0,1,2,\cdots ,0<q<p\leqq 1,$ the $(p,q)$-Bernstein operators were introduced by Mursaleen et al. [4]:

$$B_r^{p,q}(g;y)=\frac{1}{p^{\frac{r(r-1)}{2}}}\sum _{m=0}^r{\left[\begin{array}{c}r\\ m\end{array}\right]}_{p,q}{p}^{\frac{m(m-1)}{2}}{y}^k\prod _{s=0}^{r-m-1}(p^s-q^{s}y)g\left(\frac{{\left[m\right]}_{p,q}}{p^{m-r}{\left[r\right]}_{p,q}}\right),y\in [0,1],$$

(1)

where ${\left[r\right]}_{p,q}$ denotes the $(p,q)$-integer.

The $(p,q)$-analogues of exponential functions are defined in two forms as follows:

$$e_r^{p,q}\left(y\right)=\sum _{r=0}^{\infty }{p}^{\frac{r(r-1)}{2}}\frac{y^r}{{\left[r\right]}_{p,q}!},E_r^{p,q}\left(y\right)=\sum _{r=0}^{\infty }{q}^{\frac{r(r-1)}{2}}\frac{y^r}{{\left[r\right]}_{p,q}!},$$

with the property that $e_r^{p,q}\left(y\right)E_r^{p,q}(-y)=1.$ In the case of $p=1$, $e_r^{p,q}\left(y\right)$ and $E_r^{p,q}\left(y\right)$ reduce to q-analogues of exponential functions.

The Dunkl-type generalization of Szász operators [5] was introduced by Sucu [6] and the q-analogue by Ben Cheikh et al. [7]. Içöz [8] introduced the q-Dunkl analogue of Szász operators defined by:

$$D_{\eta }^q(g;y)=\frac{1}{e_{\eta }^q\left({\left[r\right]}_{q}y\right)}\sum _{m=0}^{\infty }\frac{{\left({\left[r\right]}_{q}y\right)}^m}{{\gamma }_{\eta }^q\left(m\right)}g\left(\frac{1-q^{2\eta {\theta }_m+m}}{1-q^r}\right)$$

(2)

where $\eta >-\frac{1}{2},y\geqq 0,0<q<1$, $g\in C[0,\infty )$ and $C[0,\infty )$ is the set of all continuous functions defined on $[0,\infty ).$

The $(p,q)$- and q-Dunkl analogues have been studied by several authors (see [9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]). For the most recent work on $(p,q)$-approximation, we refer to [25,26,27]. Recently. Alotaibi et al. [28] generalized the q-Dunkl analogue of Szász operators via $\left(pq\right)$-calculus as follows:

$${\mathcal{D}}_{\eta }^{p,q}(g;y)=\frac{1}{e_{\eta }^{p,q}\left({\left[r\right]}_{p,q}y\right)}\sum _{m=0}^{\infty }\frac{{\left({\left[r\right]}_{p,q}y\right)}^m}{{\gamma }_{\eta }^{p,q}\left(m\right)}{p}^{\frac{m(m-1)}{2}}g\left(\frac{p^{2\eta {\theta }_m+m}-q^{2\eta {\theta }_m+m}}{p^{m-1}(p^r-q^r)}\right)$$

(3)

where for $q\in (0,1)$, $p\in (q,1]$, and $\eta >-\frac{1}{2},$ the $(p,q)$-Dunkl analogue of exponential functions is defined by:

$$e_{\eta }^{p,q}=\sum _{r=0}^{\infty }{p}^{\frac{r(r-1)}{2}}\frac{y^r}{{\gamma }_{\eta }^{p,q}\left(r\right)},y\in [0,\infty )$$

(4)

$${\gamma }_{\eta }^{p,q}\left(r\right)=\frac{{\prod }_{i=0}^{\left[\frac{r+1}{2}\right]-1}{p}^{2\eta {(-1)}^{i+1}+1}\left({\left(p^2\right)}^i{p}^{2\eta +1}-{\left(q^2\right)}^i{q}^{2\eta +1}\right){\prod }_{j=0}^{\left[\frac{r}{2}\right]-1}{p}^{2\eta {(-1)}^j+1}\left({\left(p^2\right)}^j{p}^2-{\left(q^2\right)}^j{q}^2\right)}{{(p-q)}^r},$$

(5)

$${\gamma }_{\eta }^{p,q}(r+1)=\frac{p^{2\eta {(-1)}^{r+1}+1}\left(p^{2\eta {\theta }_{r+1}+r+1}-q^{2\eta {\theta }_{r+1}+r+1}\right)}{(p-q)}{\gamma }_{\eta }^{p,q}\left(r\right),$$

(6)

$${\theta }_r=\left\{\begin{array}{cc}0\hfill & \mathrm{for}r=2\ell ,\ell =1,2,\cdots ,n\hfill \\ 1\hfill & \mathrm{for}r=2\ell +1,\ell =1,2,\cdots ,n.\hfill \end{array}\right.$$

(7)

and $\left[\frac{r}{2}\right]$ denotes the greatest integer function; also, we have:

$${(\alpha -\beta )}_{p,q}^r=\left\{\begin{array}{cc}\prod _{j=0}^{r-1}(p^j\alpha -q^j\beta )\hfill & \mathrm{if}r=1,2,\cdots ,n\hfill \\ 1\hfill & \mathrm{if}r=0.\hfill \end{array}\right.$$

Lemma 1.

For $g\left(t\right)=1,t,t^2$

$1^{*}.$${\mathcal{D}}_{\eta }^{p,q}(1;y)=1;$

$2^{*}.$${\mathcal{D}}_{\eta }^{p,q}(t;y)=y;$

$3^{*}.$$y^2+\frac{q^{2\eta }}{{\left[r\right]}_{p,q}}{[1-2\eta ]}_{p,q}\frac{e_{\eta }^{p,q}\left(\frac{q}{p}{\left[r\right]}_{p,q}y\right)}{e_{\eta }^{p,q}\left({\left[r\right]}_{p,q}y\right)}y\leqq {\mathcal{D}}_{\eta }^{p,q}(t^2;y)\leqq y^2+\frac{1}{{\left[r\right]}_{p,q}}{[1+2\eta ]}_{p,q}y$.

2. New Operators and Estimations of Moments

In this section, we construct the $(p,q)$-variant of Szász–Kantorovich operators via Dunkl-type generalization as follows.

Definition 1.

For any $y\in [0,\infty ),g\in C[0,\infty )r\in \mathbb{N}$ and $0<q<p\leqq 1,$ we define:

$${\mathcal{K}}_{\eta }^{p,q}(g;y)=\frac{{\left[r\right]}_{p,q}}{e_{\eta }^{p,q}\left({\left[r\right]}_{p,q}y\right)}\sum _{m=0}^{\infty }\frac{{\left({\left[r\right]}_{p,q}y\right)}^m}{{\gamma }_{\eta }^{p,q}\left(m\right)}{p}^{-(m+2\eta {\theta }_m)}{p}^{\frac{m(m-1)}{2}}{\int }_{q\mathcal{A}}^{q\mathcal{A}+\mathcal{B}}g\left(\frac{t}{q{p}^{m-1}}\right)d_{p,q}t.$$

(8)

We use the following relation:

$${[m+1+2\eta {\theta }_m]}_{p,q}=q{[m+2\eta {\theta }_m]}_{p,q}+p^{m+2\eta {\theta }_m},$$

(9)

$$\mathcal{A}=\frac{{[m+2\eta {\theta }_m]}_{p,q}}{{\left[r\right]}_{p,q}},\mathcal{B}=\frac{p^{m+2\eta {\theta }_m}}{{\left[r\right]}_{p,q}}$$

(10)

where the parameter $\eta \geqq 0$.

To show the uniform convergence of operators ${\mathcal{K}}_{\eta }^{p,q}(·;·)$, we take $q=q_r,p=p_r$ with $0<q_r<1$ and $q_r<p_r\leqq 1$ such that:

$$\underset{r\to \infty }{lim}{p}_r\to 1,\underset{r\to \infty }{lim}{q}_r\to 1,\underset{r\to \infty }{lim}{p}_r^r\to u,\underset{r\to \infty }{lim}{q}_r^r\to v,(0<u,v\leqq 1).$$

(11)

For $p=1$, these operators reduce to the operators defined in [29]. For $\eta =0$, these are reduced to the $(p,q)$-variant of Kantorovich-type operators defined by [30].

Lemma 2.

Let $g\left(t\right)=g_i$ such that $g_i=t^{i-1}$ for $i=1,2,3$. Then, we have:

$\left(1\right)$${\mathcal{K}}_{\eta }^{p,q}(g_1;y)=1$

$\left(2\right)$${\mathcal{K}}_{\eta }^{p,q}(g_2;y)\leqq {\displaystyle \frac{2}{{\left[2\right]}_{p,q}}}y+{\displaystyle \frac{1}{{\left[2\right]}_{p,q}q{\left[r\right]}_{p,q}}}$

$\left(3\right)$${\mathcal{K}}_{\eta }^{p,q}(g_3;y)\leqq {\displaystyle \frac{3}{{\left[3\right]}_{p,q}}}{y}^2+{\displaystyle \frac{3}{{\left[3\right]}_{p,q}{\left[r\right]}_{p,q}}}\left({[1+2\eta ]}_{p,q}+{\displaystyle \frac{1}{q{\left[r\right]}_{p,q}}}\right)y+{\displaystyle \frac{1}{{\left[3\right]}_{p,q}{q}^2{\left[r\right]}_{p,q}^2}}.$

Proof. 

Using (9) and (10), we get:

$${\int }_{q\mathcal{A}}^{q\mathcal{A}+\mathcal{B}}f\left(\frac{t}{q{p}^{k-1}}\right)d_{p,q}t=\left\{\begin{array}{cc}\mathcal{B}\hfill & \mathrm{for}g\left(t\right)=g_1\hfill \\ \frac{\mathcal{B}}{{\left[2\right]}_{p,q}{p}^{m-1}q}\left(2q\mathcal{A}+\mathcal{B}\right)\hfill & \mathrm{for}g\left(t\right)=g_2\hfill \\ \frac{\mathcal{B}}{{\left[3\right]}_{p,q}{p}^{2(m-1)}{q}^2}\left(3q^2{\mathcal{A}}^2+3q\mathcal{A}\mathcal{B}+{\mathcal{B}}^2\right)\hfill & \mathrm{for}g\left(t\right)=g_3\hfill \end{array}\right.$$

(12)

If we take $g\left(t\right)=g_1$, then from (12), we have:

$$\begin{array}{ccc}\hfill {\mathcal{K}}_{\eta }^{p,q}(g_1;y)& =& \frac{{\left[r\right]}_{p,q}}{e_{\eta }^{p,q}\left({\left[r\right]}_{p,q}y\right)}\sum _{m=0}^{\infty }\frac{{({\left[r\right]}_{p,q}y)}^m}{{\gamma }_{\eta }^{p,q}\left(m\right)}{p}^{-(m+2\eta {\theta }_m)}{p}^{\frac{m(m-1)}{2}}{\int }_{q\mathcal{A}}^{q\mathcal{A}+\mathcal{B}}{d}_{p,q}t\hfill \\ & =& 1.\hfill \end{array}$$

For $g\left(t\right)=g_2,$ (12) implies:

$$\begin{array}{ccc}\hfill {\mathcal{K}}_{\eta }^{p,q}(g_2;y)& =& \frac{1}{{\left[2\right]}_{p,q}q{\left[r\right]}_{p,q}}\frac{1}{e_{\eta }^{p,q}\left({\left[r\right]}_{p,q}y\right)}\sum _{m=0}^{\infty }\frac{{({\left[r\right]}_{p,q}y)}^m}{{\gamma }_{\eta }^{p,q}\left(m\right)}{p}^{-(m+2\eta {\theta }_m)}{p}^{\frac{m(m-1)}{2}}\hfill \\ & \times & p^{1+2\eta {\theta }_m}\left(2q{[m+2\eta {\theta }_m]}_{p,q}+p^{m+2\eta {\theta }_m}\right)\hfill \\ & =& \frac{2}{{\left[2\right]}_{p,q}{\left[r\right]}_{p,q}}\frac{1}{e_{\eta }^{p,q}\left({\left[r\right]}_{p,q}y\right)}\sum _{m=0}^{\infty }\frac{{({\left[r\right]}_{p,q}y)}^m}{{\gamma }_{\eta }^{p,q}\left(m\right)}{p}^{\frac{(m-1)(m-2)}{2}}{[m+2\eta {\theta }_m]}_{p,q}\hfill \\ & +& \frac{1}{{\left[2\right]}_{p,q}q{\left[r\right]}_{p,q}}\frac{1}{e_{\eta }^{p,q}\left({\left[r\right]}_{p,q}y\right)}\sum _{m=0}^{\infty }\frac{{({\left[r\right]}_{p,q}y)}^m}{{\gamma }_{\eta }^{p,q}\left(m\right)}{p}^{1+2\eta {\theta }_m}\hfill \\ & =& \frac{2}{{\left[2\right]}_{p,q}}\frac{1}{e_{\eta }^{p,q}\left({\left[r\right]}_{p,q}y\right)}\sum _{m=0}^{\infty }\frac{{({\left[r\right]}_{p,q}y)}^m}{{\gamma }_{\eta }^{p,q}\left(m\right)}{p}^{\frac{m(m-1)}{2}}\left(\frac{p^{m+2\eta {\theta }_m}-q^{m+2\eta {\theta }_m}}{p^{m-1}(p^r-q^r)}\right)\hfill \\ & +& \frac{1}{{\left[2\right]}_{p,q}q{\left[r\right]}_{p,q}}\frac{1}{e_{\eta }^{p,q}\left({\left[r\right]}_{p,q}y\right)}\sum _{m=0}^{\infty }\frac{{({\left[r\right]}_{p,q}y)}^m}{{\gamma }_{\eta }^{p,q}\left(m\right)}{p}^{1+2\eta {\theta }_m}.\hfill \end{array}$$

Separating into even and odd terms, we get:

$$\begin{array}{ccc}\hfill {\mathcal{K}}_{\eta }^{p,q}(g_2;y)& =& {\displaystyle \frac{2}{{\left[2\right]}_{p,q}}}y+{\displaystyle \frac{p}{{\left[2\right]}_{p,q}q{\left[r\right]}_{p,q}}}\mathrm{for}r=0,2,4,\cdots \hfill \\ \hfill {\mathcal{K}}_{\eta }^{p,q}(g_2;y)& =& {\displaystyle \frac{2}{{\left[2\right]}_{p,q}}}y+{\displaystyle \frac{p^{1+2\eta }}{{\left[2\right]}_{p,q}q{\left[r\right]}_{p,q}}}\mathrm{for}r=1,3,5,\cdots .\hfill \end{array}$$

Since $0<q<p\leqq 1,$ $\eta \geqq 0$, and $p^{1+2\eta }\leqq 1$, we have:

$${\mathcal{K}}_{\eta }^{p,q}(g_2;y)\leqq {\displaystyle \frac{2}{{\left[2\right]}_{p,q}}}y+{\displaystyle \frac{1}{q{\left[2\right]}_{p,q}{\left[r\right]}_{p,q}}}.$$

Similarly for $g\left(t\right)=g_3,$ we have:

$$\begin{array}{ccc}\hfill {\mathcal{K}}_{\eta }^{p,q}(g_3;y)& =& \frac{3}{{\left[3\right]}_{p,q}{\left[r\right]}_{p,q}^3}\frac{1}{e_{\eta }^{p,q}\left({\left[r\right]}_{p,q}y\right)}\sum _{m=0}^{\infty }\frac{{({\left[r\right]}_{p,q}y)}^m}{{\gamma }_{\eta }^{p,q}\left(m\right)}{p}^{\frac{m(m-1)}{2}-2(m-1)}{[m+2\eta {\theta }_m]}_{p,q}^2\hfill \\ & +& \frac{3}{{\left[3\right]}_{p,q}q{\left[r\right]}_{p,q}^3}\frac{1}{e_{\eta }^{p,q}\left({\left[r\right]}_{p,q}y\right)}\sum _{m=0}^{\infty }\frac{{({\left[r\right]}_{p,q}y)}^m}{{\gamma }_{\eta }^{p,q}\left(m\right)}{p}^{m+2\eta {\theta }_m}{p}^{\frac{m(m-1)}{2}-2(m-1)}{[m+2\eta {\theta }_m]}_{p,q}\hfill \\ & +& \frac{1}{{\left[3\right]}_{p,q}{q}^2{\left[r\right]}_{p,q}^2}\frac{1}{e_{\eta }^{p,q}\left({\left[r\right]}_{p,q}y\right)}\sum _{m=0}^{\infty }\frac{{({\left[r\right]}_{p,q}y)}^m}{{\gamma }_{\eta }^{p,q}\left(m\right)}{p}^{2(m+2\eta {\theta }_m)}{p}^{\frac{m(m-1)}{2}-2(m-1)}\hfill \\ & =& \frac{3}{{\left[3\right]}_{p,q}{\left[r\right]}_{p,q}^2}\frac{1}{e_{\eta }^{p,q}\left({\left[r\right]}_{p,q}y\right)}\sum _{m=0}^{\infty }\frac{{({\left[r\right]}_{p,q}y)}^m}{{\gamma }_{\eta }^{p,q}\left(m\right)}{p}^{\frac{m(m-1)}{2}}{\left(\frac{p^{m+2\eta {\theta }_m}-q^{m+2\eta {\theta }_m}}{p^{m-1}(p^r-q^r)}\right)}^2\hfill \\ & +& \frac{3}{{\left[3\right]}_{p,q}q{\left[r\right]}_{p,q}^2}\frac{1}{e_{\eta }^{p,q}\left({\left[r\right]}_{p,q}y\right)}\sum _{m=0}^{\infty }\frac{{({\left[r\right]}_{p,q}y)}^m}{{\gamma }_{\eta }^{p,q}\left(m\right)}{p}^{\frac{m(m-1)}{2}}\left(\frac{p^{m+2\eta {\theta }_m}-q^{m+2\eta {\theta }_m}}{p^{m-1}(p^r-q^r)}\right)\hfill \\ & +& \frac{1}{{\left[3\right]}_{p,q}{q}^2{\left[r\right]}_{p,q}^2}\frac{1}{e_{\eta }^{p,q}\left({\left[r\right]}_{p,q}y\right)}\sum _{m=0}^{\infty }\frac{{({\left[r\right]}_{p,q}y)}^m}{{\gamma }_{\eta }^{p,q}\left(m\right)}{p}^{\frac{m(m-1)}{2}}{p}^{2(1+2\eta {\theta }_m)}.\hfill \end{array}$$

Hence, for $m=0,2,4,\cdots ,$ we have:

$${\mathcal{K}}_{\eta }^{p,q}(g_3;y)\leqq {\displaystyle \frac{3}{{\left[3\right]}_{p,q}}}{y}^2+{\displaystyle \frac{3}{{\left[3\right]}_{p,q}{\left[r\right]}_{p,q}}}\left({[1+2\eta ]}_{p,q}+{\displaystyle \frac{p}{q{\left[r\right]}_{p,q}}}\right)y+{\displaystyle \frac{p^2}{q^2{\left[3\right]}_{p,q}{\left[r\right]}_{p,q}^2}},$$

and for $m=1,3,5,\cdots ,$

$${\mathcal{K}}_{\eta }^{p,q}(g_3;y)\leqq {\displaystyle \frac{3}{{\left[3\right]}_{p,q}}}{y}^2+{\displaystyle \frac{3}{{\left[3\right]}_{p,q}{\left[r\right]}_{p,q}}}\left({[1+2\eta ]}_{p,q}+{\displaystyle \frac{p}{q{\left[r\right]}_{p,q}}}\right)y+{\displaystyle \frac{p^2}{{\left[3\right]}_{p,q}{q}^2{\left[r\right]}_{p,q}^2}}.$$

Therefore,

$${\mathcal{K}}_{\eta }^{p,q}(g_3;y)\leqq {\displaystyle \frac{3}{{\left[3\right]}_{p,q}}}{y}^2+{\displaystyle \frac{3}{{\left[3\right]}_{p,q}{\left[r\right]}_{p,q}}}\left({[1+2\eta ]}_{p,q}+{\displaystyle \frac{1}{q{\left[r\right]}_{p,q}}}\right)y+{\displaystyle \frac{1}{{\left[3\right]}_{p,q}{q}^2{\left[r\right]}_{p,q}^2}}.$$

This completes the proof of Lemma 2. □

Lemma 3.

Let ${\chi }_i={(t-y)}^i$ for $i=1,2.$ Then, we have:

$${\mathcal{K}}_{\eta }^{p,q}({\chi }_i;y)\leqq \left\{\begin{array}{cc}\left({\displaystyle \frac{2}{{\left[2\right]}_{p,q}}}-1\right)y+{\displaystyle \frac{1}{{\left[2\right]}_{p,q}q{\left[r\right]}_{p,q}}}\hfill & fori=1\hfill \\ \left({\displaystyle \frac{3}{{\left[3\right]}_{p,q}}}+1-{\displaystyle \frac{4}{{\left[2\right]}_{p,q}}}\right)y^2\hfill \\ +{\displaystyle \frac{1}{q{\left[r\right]}_{p,q}}}\left({\displaystyle \frac{3}{{\left[3\right]}_{p,q}}}\left({\displaystyle \frac{1}{{\left[r\right]}_{p,q}}}+q{[1+2\eta ]}_{p,q}\right)-{\displaystyle \frac{2}{{\left[2\right]}_{p,q}}}\right)y\hfill \\ +{\displaystyle \frac{1}{{\left[3\right]}_{p,q}{q}^2{\left[r\right]}_{p,q}^2}}\hfill & fori=2.\hfill \end{array}\right.$$

(13)

3. Main Results

In this section, we study the Korovkin-type approximation theorems for positive linear operators ${\mathcal{K}}_{\eta }^{p,q}(·;·)$ defined by (8). We denote the set of all bounded and continuous functions by $C_B[0,\infty )$ equipped with norm $\parallel g{\parallel }_{C_B}={sup}_{y\in [0,\infty )}\mid g\left(y\right)\mid$. We write:

$$\mathfrak{E}:=\{g\left(y\right):y\in [0,\infty ),\frac{g\left(y\right)}{1+y^2}\mathrm{is}\mathrm{convergent}\mathrm{as}y\to \infty \}.$$

Let:

$$B_{\sigma }[0,\infty )=\left\{g:\left|g\left(y\right)\right|\leqq {\mathcal{M}}_g\sigma \left(y\right)\right\},$$

$$C_{\sigma }[0,\infty )=\left\{g:g\in B_{\sigma }[0,\infty )\cap C[0,\infty )\right\},$$

$$C_{\sigma }^k[0,\infty )=\left\{g:g\in C_{\sigma }[0,\infty )\mathrm{and}\underset{y\to \infty }{lim}\frac{g\left(y\right)}{\sigma \left(y\right)}=k\right\},$$

where $\sigma \left(y\right)$ is the weight function given by $\sigma \left(y\right)=1+y^2,$ k is a constant, and ${\mathcal{M}}_g$ depends on g. $C_{\sigma }[0,\infty )$ is equipped with the norm ${\left|\right|g\left|\right|}_{\sigma }={sup}_{y\in [0,\infty )}\frac{\left|g\right(y\left)\right|}{\sigma \left(y\right)}$.

Theorem 1.

Let $q_r,p_r$ be the real numbers, with $q_r\in (0,1)$ and $p_r\in (q_r,1]$ for every integer r, satisfying $\left(q_r\right)\to 1$ and $\left(p_r\right)\to 1$ as $r\to \infty$. Then, for every $g\in C[0,\infty )\cap \mathfrak{E},$

$$\underset{r\to \infty }{lim}{\mathcal{K}}_{\eta }^{p_r,q_r}(g;y)=g\left(y\right)$$

uniformly on each compact subset of $[0,\infty )$.

Proof. 

For the proof of the uniform convergence of the operators ${\mathcal{K}}_{\eta }^{p_r,q_r}$ on each compact subset of $[0,\infty ),$ we apply the well-known Korovkin theorem [31]. It is sufficient to show that ${lim}_{r\to \infty }{\mathcal{K}}_{\eta }^{p_r,q_r}\left(g_i;y\right)=y^{i-1},$ where $g_i=t^{i-1}$ for $i=1,2,3.$

Clearly, if $q_r\to 1,$$p_r\to 1$ as $r\to \infty ,$ then $\frac{1}{{\left[r\right]}_{p_r,q_r}}\to 0,$ $\frac{r}{{\left[r\right]}_{p_r,q_r}}\to 1$. This yields that:

$$\underset{r\to \infty }{lim}{\mathcal{K}}_{\eta }^{p_r,q_r}(g_1;y)=1,\underset{r\to \infty }{lim}{\mathcal{K}}_{\eta }^{p_r,q_r}(g_2;y)=y,\underset{r\to \infty }{lim}{\mathcal{K}}_{\eta }^{p_r,q_r}(g_3;y)=y^2.$$

 □

Theorem 2.

Let $q_r,p_r$ be the real numbers, with $q_r\in (0,1)$ and $p_r\in (q_r,1]$ for every integer r, satisfying $\left(q_r\right)\to 1$ and $\left(p_r\right)\to 1$ as $r\to \infty$. Then, for every $g\in C_{\sigma }^k[0,\infty ),$ we have:

$$\underset{r\to \infty }{lim}{\left|\left|{\mathcal{K}}_{\eta }^{p_r,q_r}(g;y)-g\right|\right|}_{\sigma }=0.$$

(14)

Proof. 

Suppose $g\left(t\right)\in C_{\sigma }^k[0,\infty )$ and $g\left(t\right)=g_{\tau },$ where $g_{\tau }=t^{\tau -1}$ for $\tau =1,2,3$. Then, from the well-known Korovkin theorem, we have ${\mathcal{K}}_{\eta }^{p_r,q_r}(g_{\tau };y)\to y^{\tau -1}$ ($r\to \infty$) uniformly for each $\tau =1,2,3$. Hence, from Lemma 2, we have:

$$\underset{r\to \infty }{lim}{\left|\left|{\mathcal{K}}_{\eta }^{p_r,q_r}\left(g_1;y\right)-1\right|\right|}_{\sigma }=0.$$

(15)

For $\tau =2$,

$$\begin{array}{c}{\left|\left|{\mathcal{K}}_{\eta }^{p_r,q_r}\left(g_2;y\right)-y\right|\right|}_{\sigma }\hfill \\ =\underset{y\geqq 0}{sup}\frac{\left|{\mathcal{K}}_{\eta }^{p_r,q_r}(g_2;y)-y\right|}{1+y^2}\hfill \\ \leqq \left({\displaystyle \frac{2}{{\left[2\right]}_{p_r,q_r}}}-1\right)\underset{y\geqq 0}{sup}{\displaystyle \frac{y}{1+y}}+{\displaystyle \frac{1}{q_r{\left[2\right]}_{p_r,q_r}{\left[r\right]}_{p_r,q_r}}}\underset{y\geqq 0}{sup}{\displaystyle \frac{1}{1+y}}.\hfill \end{array}$$

Then:

$$\underset{r\to \infty }{lim}{\left|\left|{\mathcal{K}}_{\eta }^{p_r,q_r}\left(g_2;y\right)-y\right|\right|}_{\sigma }=0.$$

(16)

Similarly, if we take $\tau =3$,

$$\begin{array}{c}{\left|\left|{\mathcal{K}}_{\eta }^{p_r,q_r}\left(g_3;y\right)-y^2\right|\right|}_{\sigma }\hfill \\ =\underset{y\geqq 0}{sup}\frac{\left|{\mathcal{K}}_{\eta }^{p_r,q_r}(g_3;y)-y^2\right|}{1+y^2}\hfill \\ \leqq \left({\displaystyle \frac{3}{{\left[3\right]}_{p_r,q_r}}}-1\right)\underset{y\geqq 0}{sup}{\displaystyle \frac{y^2}{1+y^2}}\hfill \\ +{\displaystyle \frac{3}{{\left[3\right]}_{p_r,q_r}{\left[r\right]}_{p_r,q_r}}}\left({[1+2\eta ]}_{p_r,q_r}+{\displaystyle \frac{1}{q_r{\left[r\right]}_{p_r,q_r}}}\right)\underset{y\geqq 0}{sup}{\displaystyle \frac{y}{1+y^2}}\hfill \\ +{\displaystyle \frac{1}{{\left[3\right]}_{p_r,q_r}{q}_r^2{\left[r\right]}_{p_r,q_r}^2}}\underset{y\geqq 0}{sup}{\displaystyle \frac{1}{1+y^2}},\hfill \end{array}$$

$$\underset{r\to \infty }{lim}{\left|\left|{\mathcal{K}}_{\eta }^{p_r,q_r}\left(g_3;y\right)-y^2\right|\right|}_{\sigma }=0.$$

(17)

This completes the proof. □

The modulus of continuity ${\omega }_b(g;\delta )$ of the function $g\in \tilde{C}[0,\infty )$ is defined by:

$${\omega }_b(g;\delta )=\underset{\mid t-y\mid \le \delta ;}{sup}\underset{y,t\in [0,b]}{sup}\mid g\left(t\right)-g\left(y\right)\mid$$

(18)

where $\tilde{C}[0,\infty )$ denotes the space of uniformly-continuous functions on $[0,\infty ).$ It is obvious that ${lim}_{\delta \to 0+}{\omega }_b(g;\delta )=0$ and for $g\in C[0,\infty )$:

$$\mid g\left(t\right)-g\left(y\right)\mid \le \left(\frac{\mid t-y\mid }{\delta }+1\right){\omega }_b(g;\delta ).$$

(19)

Theorem 3.

Let $q_r,p_r$ be the real numbers, with $q_r\in (0,1)$ and $p_r\in (q_r,1]$ for every integer r, satisfying $\left(q_r\right)\to 1$ and $\left(p_r\right)\to 1$ as $r\to \infty$. Then, for every $g\in C_{\sigma }[0,\infty )$:

$$\left|{\mathcal{K}}_{\eta }^{p_r,q_r}(g;y)-g\left(y\right)\right|\leqq 2\left({\omega }_{b+1}(g;{\delta }_{\eta }\left(y\right))+{\mathcal{M}}_g(1+b^2){\left({\delta }_{\eta }\left(y\right)\right)}^2\right),$$

where ${\delta }_{\eta }\left(y\right)=\sqrt{{\mathcal{K}}_{\eta }^{p_r,q_r}\left({\chi }_2;y\right)}$, ${\mathcal{M}}_g$ is a constant depending only on g and ${\mathcal{K}}_{\eta }^{p_r,q_r}\left({\chi }_2;y\right)$ is defined by Lemma 3; and $[0,b+1]\subset [0,\infty ),b>0.$

Proof. 

Let $y\in [0,b]$ and $t>b+1,$ with $t>0.$ Then, for $\delta >0$, we have:

$$\mid g\left(t\right)-g\left(y\right)\mid \leqq {\omega }_{b+1}(g;\mid t-y\mid )\leqq \left(1+\frac{\mid t-y\mid }{\delta }\right){\omega }_{b+1}(g;\delta ).$$

(20)

By applying the Cauchy–Schwarz inequality and the linearity of ${\mathcal{K}}_{\eta }^{p_r,q_r}$:

$${\mathcal{K}}_{r,\eta }^{p_r,q_r}\left|g\left(t\right)-g\left(y\right);y\right|\leqq \left({\left(1+\frac{1}{\delta }{\mathcal{K}}_{\eta }^{p_r,q_r}\left({(t-y)}^2;y\right)\right)}^{\frac{1}{2}}\right){\omega }_{b+1}(g;\delta ).$$

(21)

For $t-y>1,$ we have:

$$\begin{array}{c}\left|g\left(t\right)-g\left(y\right)\right|\hfill \\ \leqq {\mathcal{M}}_g\left(2+y^2+t^2\right)\hfill \\ \leqq {\mathcal{M}}_g\left(2+3y^2+2{(t-y)}^2\right)\leqq 2{\mathcal{M}}_g(1+b^2){(t-y)}^2\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\mathcal{K}}_{\eta }^{p_r,q_r}\left(\left|g\left(t\right)-g\left(y\right)\right|;\right)& \leqq 2{\mathcal{M}}_g(1+b^2){\mathcal{K}}_{\eta }^{p_r,q_r}\left({(t-y)}^2;y\right).\hfill \end{array}$$

(22)

From (21) and (22), we easily see that:

$$\begin{array}{cc}\hfill \left|{\mathcal{K}}_{\eta }^{p_r,q_r}(g;y)-g\left(y\right)\right|& \\ & \leqq {\mathcal{K}}_{\eta }^{p_r,q_r}\left|g\left(t\right)-g\left(y\right);y\right|\hfill \\ & \leqq \left({\left(1+\frac{1}{\delta }{\mathcal{K}}_{\eta }^{p_r,q_r}\left({(t-y)}^2;y\right)\right)}^{\frac{1}{2}}\right){\omega }_{b+1}(g;\delta )\hfill \\ & +2{\mathcal{M}}_g(1+b^2){\mathcal{K}}_{\eta }^{p_r,q_r}\left({(t-y)}^2;y\right)\hfill \\ & ={\left(1+\frac{1}{\delta }{\mathcal{K}}_{\eta }^{p_r,q_r}\left({\chi }_2;y\right)\right)}^{\frac{1}{2}}{\omega }_{b+1}(g;\delta )\hfill \\ & +2{\mathcal{M}}_g(1+b^2){\mathcal{K}}_{\eta }^{p_r,q_r}\left({\chi }_2;y\right)\hfill \end{array}$$

If we choose $\delta ={\delta }_{\eta }\left(y\right)=\sqrt{{\mathcal{K}}_{\eta }^{p_r,q_r}\left({\chi }_2;y\right)}$, then we get our result. □

For any $g\in C[0,\infty ],\mathcal{L}>0,0<\nu \le 1$ and ${\gamma }_1,{\gamma }_2\in [0,\infty )$, we recall that:

$$Li{p}_{\mathcal{L}}\left(\nu \right)=\left\{g:\mid g\left({\gamma }_1\right)-g\left({\gamma }_2\right)\mid \le \mathcal{L}\mid {\gamma }_1-{\gamma }_2{\mid }^{\nu }\right\}.$$

(23)

Theorem 4.

Let $q_r,p_r$ be the real numbers, with $q_r\in (0,1)$ and $p_r\in (q_r,1]$ for every integer r, satisfying $\left(q_r\right)\to 1$ and $\left(p_r\right)\to 1$ as $r\to \infty$. Then, for each $g\in Li{p}_{\mathcal{L}}\left(\nu \right),$ we have:

$$\left|{\mathcal{K}}_{\eta }^{p_r,q_r}(g;y)-g\left(y\right)\right|\leqq \mathcal{L}{\left({\delta }_{\eta }\left(y\right)\right)}^{\nu },$$

where ${\delta }_{\eta }\left(y\right)$ is defied by Theorem 3.

Proof. 

Using Theorem 4, (23), and the well-known Hölder’s inequality, we get:

$$\begin{array}{ccc}\hfill \left|{\mathcal{K}}_{\eta }^{p_r,q_r}(g;y)-g\left(y\right)\right|& \leqq & \left|{\mathcal{K}}_{r,\eta }^{p_r,q_r}(g\left(t\right)-g\left(y\right);y)\right|\hfill \\ & \leqq & {\mathcal{K}}_{\eta }^{p_r,q_r}\left(\left|g\left(t\right)-g\left(y\right)\right|;y\right)\hfill \\ & \leqq & \mid \mathcal{L}{\mathcal{K}}_{\eta }^{p_r,q_r}\left({\left|t-y\right|}^{\nu };y\right)\hfill \\ & \leqq & \mathcal{L}{\left({\mathcal{K}}_{\eta }^{p_r,q_r}(g_1;y)\right)}^{\frac{2-\nu }{2}}{\left({\mathcal{K}}_{\eta }^{p_r,q_r}({\left|t-y\right|}^2;y)\right)}^{\frac{\nu }{2}}\hfill \\ & =& \mathcal{L}{\left({\mathcal{K}}_{\eta }^{p_r,q_r}({\chi }_2;y)\right)}^{\frac{\nu }{2}}.\hfill \end{array}$$

This completes the proof of the theorem. □

We denote:

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

(24)

$${\left|\right|\psi \left|\right|}_{C_B^2\left({\mathbb{R}}^{+}\right)}={\left|\right|\psi \left|\right|}_{C_B[0,\infty )}+{\left|\left|{\psi }^{\prime }\right|\right|}_{C_B[0,\infty )}+{\left|\left|{\psi }^{″}\right|\right|}_{C_B[0,\infty )},$$

(25)

$${\left|\right|\psi \left|\right|}_{C_B[0,\infty )}=\underset{y\in [0,\infty )}{sup}\left|\psi \left(y\right)\right|.$$

(26)

Theorem 5.

Let $q_r,p_r$ be the real numbers, with $q_r\in (0,1)$ and $p_r\in (q_r,1]$ for every integer r, satisfying $\left(q_r\right)\to 1$ and $\left(p_r\right)\to 1$ as $r\to \infty$. Then:

$$\left|{\mathcal{K}}_{\eta }^{p_r,q_r}(\psi ;y)-\psi \left(y\right)\right|\leqq {{\rm Y}}_{\eta }{\left(y\right)\left|\right|\psi \left|\right|}_{C_B^2[0,\infty )},$$

(27)

where ${{\rm Y}}_{\eta }\left(y\right)={\delta }_n\left(y\right)\left(1+\frac{{\delta }_{\eta }\left(y\right)}{2}\right)$ and ${\delta }_{\eta }\left(y\right)$ is defined by Theorem 3.

Proof. 

From the Taylor series expansion for any $\psi \in C_B^2[0,\infty ),$ we have:

$$\psi \left(t\right)=\psi \left(y\right)+{\psi }^{\prime }\left(y\right)(t-y)+{\psi }^{″}\left(\phi \right)\frac{{(t-y)}^2}{2}\mathrm{for}\phi \in (y,t),$$

$$\left|\psi \left(t\right)-\psi \left(y\right)\right|\leqq \mathfrak{P}\mid t-y\mid +\frac{1}{2}\mathfrak{Q}{(t-y)}^2,$$

where:

$$\mathfrak{P}=\underset{y[0,\infty )}{sup}\left|{\psi }^{\prime }\left(y\right)\right|=\left|\right|{\psi }^{\prime }{\left|\right|}_{C_B[0,\infty )}\leqq {\left|\right|\psi \left|\right|}_{C_B^2[0,\infty )},$$

$$\mathfrak{Q}=\underset{y[0,\infty )}{sup}\left|{\psi }^{″}\left(y\right)\right|=\left|\right|{\psi }^{″}{\left|\right|}_{C_B[0,\infty )}\leqq {\left|\right|\psi \left|\right|}_{C_B^2[0,\infty )}.$$

Therefore,

$$\left|\psi \left(t\right)-\psi \left(y\right)\right|\leqq \left(\mid t-y\mid +\frac{1}{2}{(t-y)}^2\right){\left|\right|\psi \left|\right|}_{C_B^2[0,\infty )}.$$

By applying the linearity of ${\mathcal{K}}_{\eta }^{p_r,q_r},$ we get:

$$\begin{array}{c}\left|{\mathcal{K}}_{\eta }^{p_r,q_r}(\psi ;y)-\psi \left(y\right)\right|\hfill \\ \leqq \left({\mathcal{K}}_{\eta }^{p_r,q_r}(\mid t-y\mid ;y)+\frac{1}{2}{\mathcal{K}}_{\eta }^{p_r,q_r}\left({(t-y)}^2;y\right)\right){\left|\right|\psi \left|\right|}_{C_B^2[0,\infty )}\hfill \\ \leqq \left({\left({\mathcal{K}}_{\eta }^{p_r,q_r}\left({\chi }_2;y\right)\right)}^{\frac{1}{2}}+\frac{1}{2}{\mathcal{K}}_{\eta }^{p_r,q_r}\left({\chi }_2;y\right)\right){\left|\right|\psi \left|\right|}_{C_B^2[0,\infty )}\hfill \\ =\left({\delta }_{\eta }\left(y\right)+\frac{{\left({\delta }_{\eta }\left(y\right)\right)}^2}{2}\right){\left|\right|\psi \left|\right|}_{C_B^2[0,\infty )}.\hfill \end{array}$$

This completes the proof of the theorem. □

Peetre’s K-functional $K_2(g;\delta )$ for $\delta >0$ (see [32]) is defined by:

$$K_2(g;\delta )=\underset{y\in [0,\infty )}{inf}\left\{\left(\delta {\left|\left|{\psi }^{″}+\left|\right|g-{\psi \left|\right|}_{C_B[0,\infty )}\right|\right|}_{C_B[0,\infty )}\right)\right\}$$

(28)

for all $\psi \in C_B^2[0,\infty ).$

For a given positive constant $\mathfrak{L}>0$:

$$K_2(g;\delta )\leqq \mathfrak{L}{\omega }_2(g;{\delta }^{\frac{1}{2}}),$$

where the second-order modulus of continuity denoted by ${\omega }_2(g;\delta )$ is defined as:

$${\omega }_2(g;\delta )=\underset{0<h<\delta }{sup},\underset{y\in [0,\infty )}{sup}|g\left(y\right)+g(y+2h)-2g(y+h)|.$$

(29)

Theorem 6.

Let $q_r,p_r$ be the real numbers, with $q_r\in (0,1)$ and $p_r\in (q_r,1]$ for every integer r, satisfying $\left(q_r\right)\to 1$ and $\left(p_r\right)\to 1$ as $r\to \infty$. Then, for all $g\in C_B[0,\infty ),$ we have:

$$\begin{array}{c}\left|{\mathcal{K}}_{\eta }^{p_r,q_r}(g;y)-g\left(y\right)\right|\hfill \\ \leqq 2\mathcal{A}\{{\omega }_2\left(g;\sqrt{\frac{{{\rm Y}}_{\eta }\left(y\right)}{2}}\right)+min\left(1;\frac{{{\rm Y}}_{\eta }\left(y\right)}{2}\right){\left|\right|g\left|\right|}_{C_B[0,\infty )}\},\hfill \end{array}$$

where $\mathcal{A}$ is a positive constant and ${{\rm Y}}_{\eta }\left(y\right)$ is given in Theorem 5.

Proof. 

We take $\psi \in C_B^2[0,\infty )$ and apply Theorem (5). Thus:

$$\begin{array}{cc}\hfill \left|{\mathcal{K}}_{\eta }^{p_r,q_r}(g;y)-g\left(y\right)\right|& \leqq \left|{\mathcal{K}}_{\eta }^{p_r,q_r}(g-\psi ;y)\right|+\left|{\mathcal{K}}_{\eta }^{p_r,q_r}(\psi ;y)-\psi \left(y\right)\right|+|g\left(y\right)-\psi \left(y\right)|\hfill \\ & \leqq 2\left|\right|g-{\psi \left|\right|}_{C_B[0,\infty )}+{{\rm Y}}_{\eta }{\left(y\right)\left|\right|\psi \left|\right|}_{C_B^2[0,\infty )}\hfill \\ & =2\left(\left|\right|g-{\psi \left|\right|}_{C_B[0,\infty )}+\frac{{{\rm Y}}_{\eta }\left(y\right)}{2}{\left|\right|\psi \left|\right|}_{C_B^2[0,\infty )}\right).\hfill \end{array}$$

By taking the infimum over all $\psi \in C_B^2[0,\infty )$ and using (28), we get:

$$\left|{\mathcal{K}}_{\eta }^{p_r,q_r}(g;y)-g\left(y\right)\right|\leqq 2K_2\left(g;\frac{{{\rm Y}}_{\eta }\left(y\right)}{2}\right).$$

Now, from [33] for all $g\in C_B[0,\infty ),$ we have the relation:

$$K_2(g;\delta )\leqq \mathcal{A}\{min(1;\delta )+{\omega }_2(g;\sqrt{\delta }){\left|\right|g\left|\right|}_{C_B[0,\infty )}\},$$

where $\mathcal{A}>0$ is an absolute constant. If we choose $\delta =\frac{{{\rm Y}}_{\eta }\left(y\right)}{2}$, then we get the desired result. □

4. Conclusions

In this paper, we have studied the approximation results via Dunkl generalization of the Szász–Kantorovich operators in $(p,q)$-calculus. These types of modifications enable us to generalize error estimation rather than the classical and q-calculus on the interval $[0,\infty )$ obtained in [29]. We have also proven the Korovkin-type results and obtained the convergence of our operators in weighted space by the modulus of continuity, Lipschitz class, and Peetre’s K-functionals. We have a more generalized version of the operators [29,30], and if we take $\eta =0$ in (8), then the operators ${\mathcal{K}}_{\eta }^{p,q}$ reduce to the operators defined by [30].