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Dunkl-type generalization of the second kind beta operators via \((p,q)\)-calculus


The main purpose of this research article is to construct a Dunkl extension of \((p,q)\)-variant of Szász–Beta operators of the second kind by applying a new parameter. We obtain Korovkin-type approximation theorems, local approximations, and weighted approximations. Further, we study the rate of convergence by using the modulus of continuity, Lipschitz class and Peetre’s K-functionals.

1 Introduction and preliminaries

The q-analogues of Bernstein operators were independently given by Lupaş [25] and Phillips [42]. Consequently, Mursaleen et al. [33] applied the \((p,q)\)-integers and studied the approximation properties of Bernstein operators. Recently, a Dunkl-type generalization of Szász operators [47] via post-quantum calculus was studied by Alotaibi et al. [10]. For more details and research motivation in Dunkl-type generalizations, we mention here some research articles [13, 27, 34, 35, 3739, 45, 46]. Let \([s]_{p,q}\) be the \((p,q)\)-integer defined as

$$\begin{aligned}& [ s ] _{p,q}=p^{s-1}+qp^{s-3}+\cdots +q^{s-1}= \textstyle\begin{cases} \frac{p^{s}-q^{s}}{p-q} & (p\neq q\neq 1), \\ \frac{1-q^{s}}{1-q} & (p=1), \\ s & (p=q=1),\end{cases}\displaystyle \\& (au+bv)_{p,q}^{s}:=\sum_{\ell =0}^{s}p^{ \frac{(s-\ell )(s-\ell -1)}{2}}q^{\frac{\ell (\ell -1)}{2}} \begin{bmatrix} s \\ \ell \end{bmatrix} _{p,q}a^{s-\ell }b^{\ell }u^{s-\ell }v^{\ell }, \\& (1-u)_{p,q}^{s}=(1-u) (p-qu) \bigl(p^{2}-q^{2}u \bigr)\cdots \bigl(p^{s-1}-q^{s-1}u\bigr), \\& (u-y)_{p,q}^{s}=\textstyle\begin{cases} \prod_{j=0}^{s-1}(p^{j}u-q^{j}y) & \text{if }s\in \mathbb{N}, \\ 1 & \text{if }s=0.\end{cases}\displaystyle \end{aligned}$$

The \((p,q)\)-power basis is explained as

$$ (u\oplus v)_{p,q}^{s}=(u+v) (pu+qv) \bigl(p^{2}u+q^{2}v \bigr)\cdots \bigl(p^{s-1}u+q^{s-1}v\bigr). $$

Furthermore, the \((p,q)\)-analogues of the exponential function are defined by

$$ e_{p,q}(u)=\sum_{\ell =0}^{\infty }p^{\frac{\ell (\ell -1)}{2}} \frac{u^{\ell }}{[\ell ]_{p,q}!},\qquad E_{p,q}(u)=\sum _{\ell =0}^{ \infty }q^{\frac{\ell (\ell -1)}{2}}\frac{u^{\ell }}{[\ell ]_{p,q}!}. $$

Moreover, the \((p,q)\)-Dunkl analogue of the exponential function is defined by

$$\begin{aligned}& e_{\tau ,p,q}(u)=\sum_{\ell =0}^{\infty }p^{\frac{\ell (\ell -1)}{2}} \frac{u^{\ell }}{\gamma _{\tau ,p,q}(\ell )} , \end{aligned}$$
$$\begin{aligned}& \gamma _{\tau ,p,q}(\ell ) \\& \quad = \frac{\prod_{i=0}^{[\frac{\ell +1}{2}]-1}p^{2\tau (-1)^{i+1}+1} ( (p^{2})^{i}p^{2\tau +1}-(q^{2})^{i}q^{2\tau +1} ) \prod_{j=0}^{[\frac{\ell }{2}]-1}p^{2\tau (-1)^{j}+1} ( (p^{2})^{j}p^{2}-(q^{2})^{j}q^{2} ) }{(p-q)^{\ell }}. \end{aligned}$$

And a recursion identity is defined as

$$ \gamma _{\tau ,p,q}(\ell +1)= \frac{p^{2\tau (-1)^{\ell +1}+1}({p^{2\tau \theta _{\ell +1}+\ell +1}-q^{2\tau \theta _{\ell +1}+\ell +1}})}{(p-q)}\gamma _{\tau ,p,q}(\ell ), $$


$$ \theta _{\ell }= \textstyle\begin{cases} 0 & \text{for }\ell =2m, m=0,1,2,\dots , \\ 1 & \text{for }\ell =2m+1, m=0,1,2,\dots \end{cases} $$

For \(m=0,1,2,\dots s\), the number \([\frac{m}{2}]\) denotes the greatest integer function evaluated at \(m/2\).

In our demonstration, we let \(u\geq 0\) and \(C[0,\infty )\) be the class of all continuous functions on \([0,\infty )\). Recent investigation in [10, 38] defined the \((p,q)\)-Dunkl analogue of Szász operators by

$$ D_{s,p,q}(f;u)=\frac{1}{e_{\tau ,p,q}([s]_{p,q}u)}\sum_{\ell =0}^{ \infty }\frac{([s]_{p,q}u)^{\ell }}{\gamma _{\tau ,p,q}(\ell )}p^{ \frac{\ell (\ell -1)}{2}}f \biggl( \frac{p^{\ell +2\tau \theta _{\ell }}-q^{\ell +2\tau \theta _{\ell }}}{p^{\ell -1}(p^{s}-q^{s})} \biggr) . $$

2 Operators and basic estimates

In this section we construct a class of \((p,q)\)-variant of Szász–Beta operators of the second kind generated by an exponential function via Dunkl generalization in Definition 2.1. Such operators are a generalized version of the operators studied in [7, 22, 28, 29, 31, 36, 45].

Definition 2.1

Let \(f\in C_{\zeta }[0,\infty )=\{ f(t):f(t)=O(t^{\zeta }), t \rightarrow \infty , f\in C[0,\infty )\}\) and consider \(u\geq 0\), \(\zeta >s\), and \(s \in \mathbb{N}\). Then for all \(0< q< p\leq 1\), \(\tau >-\frac{1}{2}\), and \(\theta _{\ell }\) given by (1.5), we define

$$ \mathcal{P}_{s,p,q}^{\tau }(f;u)=\sum _{\ell =0}^{\infty } \mathcal{Q}_{s,p,q}(u) \frac{1}{\mathcal{B}_{p,q}(\ell +2\tau \theta _{\ell }+1,s)}\int _{0}^{\infty } \frac{t^{\ell +2\tau \theta _{\ell }}}{(1\oplus pt)_{p,q}^{\ell +2\tau \theta _{\ell }+s+1}}f(t) \,\mathrm{d}_{p,q}t, $$


$$ \mathcal{Q}_{s,p,q}(u)=\frac{1}{e_{\tau ,p,q}([s]_{p,q}u)} \frac{([s]_{p,q}u)^{\ell }}{\gamma _{\tau ,p,q}(\ell )}p^{ \frac{\ell (\ell -1)}{2}}, $$

and \(\mathcal{B}_{p,q}(\ell +2\tau \theta _{\ell }+1,s)\) is the Beta function of the second kind in post-quantum calculus defined by

$$ \mathcal{B}_{p,q}(\alpha ,\beta )= \int _{0}^{\infty } \frac{t^{\alpha -1}}{(1\oplus pt)_{p,q}^{\alpha +\beta }} \,\mathrm{d}_{p,q}t,\quad \alpha , \beta \in \mathbb{N,} $$

where a formula for the \((p,q)\)-Beta function is given by

$$ \mathcal{B}_{p,q}(\alpha ,\beta ) = \frac{[\alpha -1]_{p,q}}{p^{\alpha -1} [\beta ]_{p,q}} \mathcal{B}_{p,q}(\alpha -1,\beta +1), \quad \alpha , \beta \in \mathbb{N}. $$

Moreover, to obtain the basic estimates here, we use the following relations:

$$\begin{aligned}& {}[ \ell +1+2\tau \theta _{\ell }]_{p,q}=q[ \ell +2\tau \theta _{ \ell }]_{p,q}+p^{\ell +2\tau \theta _{\ell }}, \end{aligned}$$
$$\begin{aligned}& {}[ \ell +2+2\tau \theta _{\ell }]_{p,q}=q^{2}[ \ell +2\tau \theta _{ \ell }]_{p,q}+(p+q)p^{\ell +2\tau \theta _{\ell }}. \end{aligned}$$

For more related results on \((p,q)\)-analogues, we refer to [16, 8, 9, 11, 1421, 26, 30, 43, 44, 48] and also see [12, 32, 40], for example, if \(p=1\), the operators \(\mathcal{P}_{s,p,q}^{\tau }\) reduce to those considered recently (see [45]). We have the following inequalities.

Lemma 2.2

Let \(f(t)=1,t,t^{2}\). Then the operators \(\mathcal{P}_{s,p,q}^{\tau }(\,\cdot \,;\,\cdot \,)\) defined by (2.1) satisfy \(\mathcal{P}_{s,p,q}^{\tau }(1;u)=1\), and the following inequalities hold:

$$ \mathcal{P}_{s,p,q}^{\tau }(f;u)\leq \textstyle\begin{cases} \frac{{}[ s]_{p,q}}{[s-1]_{p,q}}u+\frac{1}{[s-1]_{p,q}} \quad \textit{for } f(t)=t, \\ \frac{{}[ s]_{p,q}^{2}}{[s-1]_{p,q}[s-2]_{p,q}}u^{2}+ \frac{[s]_{p,q}}{[s-1]_{p,q}[s-2]_{p,q}} ( 1+[2]_{p,q}+[1+2\tau ]_{p,q} ) u \\ \quad {}+\frac{[2]_{p,q}}{[s-1]_{p,q}[s-2]_{p,q}} \quad \textit{for } f(t)=t^{2},\end{cases} $$


$$ \mathcal{P}_{s,p,q}^{\tau }(f;u)\geq \textstyle\begin{cases} \frac{q[s]_{p,q}}{[s-1]_{p,q}}u+\frac{1}{[s-1]_{p,q}} \quad \textit{for } f(t)=t, \\ \frac{q^{3}[s]_{p,q}^{2}}{[s-1]_{p,q}[s-2]_{p,q}}u^{2} & \\ \quad {}+\frac{q[s]_{p,q}}{[s-1]_{p,q}[s-2]_{p,q}} (q+[2]_{p,q}+q^{2+2 \tau }[1-2\tau ]_{p,q} \frac{e_{\tau ,p,q} ( \frac{q}{p}[s]_{p,q}u ) }{e_{\tau ,p,q}([s]_{p,q}u)} )u & \\ \quad {}+\frac{[2]_{p,q}}{[s-1]_{p,q}[s-2]_{p,q}} \quad \textit{for } f(t)=t^{2}.\end{cases} $$


To prove the results of this lemma, we use (2.2)–(2.5). Take \(f(t)=1\). Then

$$\begin{aligned} \mathcal{P}_{s,p,q}^{\tau }(1;u) =&\sum _{\ell =0}^{\infty } \mathcal{Q}_{s,p,q}(u) \frac{1}{\mathcal{B}_{p,q}(\ell +2\tau \theta _{\ell }+1,s)}\int _{0}^{\infty } \frac{t^{\ell +2\tau \theta _{\ell }}}{(1\oplus pt)_{p,q}^{\ell +2\tau \theta _{\ell }+s+1}} \,\mathrm{d}_{p,q}t \\ =&\sum_{\ell =0}^{\infty }\mathcal{Q}_{s,p,q}(u) \frac{\mathcal{B}_{p,q}(\ell +2\tau \theta _{\ell }+1,s)}{\mathcal{B}_{p,q}(\ell +2\tau \theta _{\ell }+1,s)} =1. \end{aligned}$$

If \(f(t)=t\), then

$$\begin{aligned} \mathcal{P}_{s,p,q}^{\tau }(t;u) =&\sum _{\ell =0}^{\infty } \mathcal{Q}_{s,p,q}(u) \frac{1}{\mathcal{B}_{p,q}(\ell +2\tau \theta _{\ell }+1,s)}\int _{0}^{\infty } \frac{t^{\ell +2\tau \theta _{\ell }+1}}{(1\oplus pt)_{p,q}^{\ell +2\tau \theta _{\ell }+s+1}} \,\mathrm{d}_{p,q}t \\ =&\sum_{\ell =0}^{\infty }\mathcal{Q}_{s,p,q}(u) \frac{\mathcal{B}_{p,q}(\ell +2\tau \theta _{\ell }+2,s-1)}{\mathcal{B}_{p,q}(\ell +2\tau \theta _{\ell }+1,s)} \\ =&\frac{q}{[s-1]_{p,q}}\sum_{\ell =0}^{\infty } \mathcal{Q}_{s,p,q}(u) \frac{1}{p^{\ell +2\tau \theta _{\ell }+1}}[\ell +2\tau \theta _{\ell }]_{p,q}+\frac{1}{p[s-1]_{p,q}} \\ =&\frac{1}{p[s-1]_{p,q}}+\frac{q[s]_{p,q}}{p^{2}[s-1]_{p,q}}\sum_{ \ell =0}^{\infty } \mathcal{Q}_{s,p,q}(u) \biggl( \frac{p^{2\ell +2\tau \theta _{2\ell }}-q^{2\ell +2\tau \theta _{2\ell }}}{p^{2\ell -1}(p^{s}-q^{s})} \biggr) \\ &{}+\frac{q[s]_{p,q}}{p^{2+2\tau }[s-1]_{p,q}}\sum_{\ell =0}^{\infty }\mathcal{Q}_{s,p,q}(u) \biggl( \frac{p^{2\ell +1+2\tau \theta _{2\ell +1}}-q^{2\ell +1+2\tau \theta _{2\ell +1}}}{p^{2\ell }(p^{s}-q^{s})} \biggr). \end{aligned}$$

Clearly, we have

$$\begin{aligned} \mathcal{P}_{s,p,q}^{\tau }(t;u) \geq &\frac{1}{[s-1]_{p,q}}+ \frac{q[s]_{p,q}}{[s-1]_{p,q}}\sum_{\ell =0}^{\infty } \mathcal{Q}_{s,p,q}(u) \biggl(\frac{p^{\ell +2\tau \theta _{\ell }}-q^{\ell +2\tau \theta _{\ell }}}{p^{\ell -1}(p^{s}-q^{s})} \biggr) \\ =&\frac{1}{[s-1]_{p,q}}+\frac{q[s]_{p,q}}{[s-1]_{p,q}}D_{s,p,q}(t;u) \\ =&\frac{1}{[s-1]_{p,q}}+\frac{q[s]_{p,q}}{[s-1]_{p,q}}u \end{aligned}$$


$$ \mathcal{P}_{s,p,q}^{\ell ,\tau }(t;u)\leq \frac{1}{[s-1]_{p,q}}+ \frac{[s]_{p,q}}{[s-1]_{p,q}}u. $$

Similarly, for \(f(t)=t^{2}\), we have

$$\begin{aligned} \mathcal{P}_{s,p,q}^{\tau }\bigl(t^{2};u\bigr) =& \sum_{\ell =0}^{\infty } \mathcal{Q}_{s,p,q}(u) \frac{1}{\mathcal{B}_{p,q}(\ell +2\tau \theta _{\ell }+1,s)}\int _{0}^{\infty } \frac{t^{\ell +2\tau \theta _{\ell }+2}}{(1\oplus pt)_{p,q}^{\ell +2\tau \theta _{\ell }+s+1}} \,\mathrm{d}_{p,q}t \\ =&\sum_{\ell =0}^{\infty }\mathcal{Q}_{s,p,q}(u) \frac{\mathcal{B}_{p,q}(\ell +2\tau \theta _{\ell }+3,s-2)}{\mathcal{B}_{p,q}(\ell +2\tau \theta _{\ell }+1,s)} \\ =&\sum_{\ell =0}^{\infty }\mathcal{Q}_{s,p,q}(u) \frac{\mathcal{B}_{p,q}(\ell +2\tau \theta _{\ell }+3,s-2)}{\mathcal{B}_{p,q}(\ell +2\tau \theta _{\ell }+1,s)} \\ =&\frac{1}{[s-1]_{p,q}[s-2]_{p,q}}\\ &{}\times\sum_{\ell =0}^{\infty } \mathcal{Q}_{s,p,q}(u)\frac{1}{p^{3+2\ell +4\tau \theta _{\ell }+1}}[\ell +2 \tau \theta _{\ell }+1]_{p,q}[\ell +2\tau \theta _{\ell }+2]_{p,q} \\ =&\frac{q^{3}[s]_{p,q}^{2}}{[s-1]_{p,q}[s-2]_{p,q}}\sum_{\ell =0}^{ \infty }\mathcal{Q}_{s,p,q}(u)\frac{1}{p^{5+4\tau \theta _{\ell }}} \biggl( \frac{p^{\ell +2\tau \theta _{\ell }}-q^{\ell +2\tau \theta _{\ell }}}{p^{\ell -1}(p^{s}-q^{s})} \biggr)^{2} \\ &{}+\frac{q(p+2q)[s]_{p,q}}{[s-1]_{p,q}[s-2]_{p,q}}\sum_{\ell =0}^{ \infty }\mathcal{Q}_{s,p,q}(u)\frac{1}{p^{4+2\tau \theta _{\ell }}} \biggl( \frac{p^{\ell +2\tau \theta _{\ell }}-q^{\ell +2\tau \theta _{\ell }}}{p^{\ell -1}(p^{s}-q^{s})} \biggr) \\ &{}+\frac{(p+q)}{p^{3}[s-1]_{p,q}[s-2]_{p,q}}\sum_{\ell =0}^{\infty } \mathcal{Q}_{s,p,q}(u). \end{aligned}$$

Now by separating the even and odd terms and applying \(\theta _{\ell }\) from (1.5), i.e., taking \(\ell =2m\) and \(\ell =2m+1\) for all \(m=0,1,2,\dots \), we have

$$\begin{aligned} \mathcal{P}_{s,p,q}^{\tau }\bigl(t^{2};u\bigr) \geq &\frac{q^{3}[s]_{p,q}^{2}}{[s-1]_{p,q}[s-2]_{p,q}}\sum_{\ell =0}^{\infty } \mathcal{Q}_{s,p,q}(u) \biggl(\frac{p^{\ell +2\tau \theta _{\ell }}-q^{\ell +2\tau \theta _{\ell }}}{p^{\ell -1}(p^{s}-q^{s})} \biggr)^{2} \\ &{}+\frac{q(q+[2]_{p,q})[s]_{p,q}}{[s-1]_{p,q}[s-2]_{p,q}}\sum_{\ell =0}^{ \infty } \mathcal{Q}_{s,p,q}(u) \biggl( \frac{p^{\ell +2\tau \theta _{\ell }}-q^{\ell +2\tau \theta _{\ell }}}{p^{\ell -1}(p^{s}-q^{s})} \biggr) \\ &{}+\frac{[2]_{p,q}}{[s-1]_{p,q}[s-2]_{p,q}} \\ =&\frac{q^{3}[s]_{p,q}^{2}}{[s-1]_{p,q}[s-2]_{p,q}}D_{s,p,q}\bigl(t^{2};u\bigr)+ \frac{q(q+[2]_{p,q})[s]_{p,q}}{[s-1]_{p,q}[s-2]_{p,q}}D_{s,p,q}(t;u) \\ &{}+\frac{[2]_{p,q}}{[s-1]_{p,q}[s-2]_{p,q}}. \end{aligned}$$


$$\begin{aligned} \mathcal{P}_{s,p,q}^{\tau }\bigl(t^{2};u\bigr) \leq &\frac{[s]_{p,q}^{2}}{[s-1]_{p,q}[s-2]_{p,q}}D_{s,p,q}\bigl(t^{2};u\bigr)+ \frac{(1+[2]_{p,q})[s]_{p,q}}{[s-1]_{p,q}[s-2]_{p,q}}D_{s,p,q}(t;u) \\ &{}+\frac{[2]_{p,q}}{[s-1]_{p,q}[s-2]_{p,q}}. \end{aligned}$$

This completes the proof of Lemma 2.2. □

Lemma 2.3

Let \(\Phi _{j}=(t-u)^{j}\) for \(j=1,2\), then we have following inequalities:

$$\begin{aligned} 1.\quad \mathcal{P}_{s,p,q}^{\tau }(\Phi _{1};u) \leq & \biggl( \frac{{}[ s]_{p,q}}{[s-1]_{p,q}}-1 \biggr) u+\frac{1}{[s-1]_{p,q}}, \quad \textit{for } s>1, s \in \mathbb{N,} \\ 2.\quad \mathcal{P}_{s,p,q}^{\tau }(\Phi _{2};u) \leq & {\biggl(} \frac{{}[ s]_{p,q}^{2}}{[s-1]_{p,q}[s-2]_{p,q}}- \frac{2[s]_{p,q}}{[s-1]_{p,q}}+1 { \biggr)} u^{2} \\ &{}+\frac{1}{[s-1]_{p,q}} \biggl(\frac{[s]_{p,q}}{[s-2]_{p,q}} \bigl(1+[2]_{p,q}+[1+2 \tau ]_{p,q} \bigr)-2 \biggr)u \\ &{}+\frac{[2]_{p,q}}{[s-1]_{p,q}[s-2]_{p,q}}, \quad \textit{for } s>2, s \in \mathbb{N}. \end{aligned}$$

3 Approximation results

Let us denote by \(C_{B}[0,\infty )\) the set of all bounded and continuous functions defined on \([0,\infty )\), equipped with the norm \(\| f\| _{C_{B}}=\sup_{u\geq 0}| f(u)| \). We write

$$\begin{aligned}& \mathcal{L}:=\biggl\{ f:\lim_{u\rightarrow \infty }\frac{f(u)}{1+u^{2}}\text{ exits}\biggr\} , \\& B_{\sigma }[0,\infty ):= \bigl\{ f: \bigl\vert f(u) \bigr\vert \leq \mathcal{M}_{f}\sigma (u) \bigr\} , \end{aligned}$$

where \(\mathcal{M}_{f}\) is a constant depending on f, and σ is the weight function with \(\sigma (u)=1+u^{2}\). Moreover,

$$\begin{aligned}& C_{\sigma }[0,\infty ):=B_{\sigma }[0,\infty )\cap C[0,\infty ), \\& C_{\sigma }^{k}[0,\infty ):= \biggl\{ f:f\in C_{\sigma }[0,\infty ) \text{ and }\lim _{u\rightarrow \infty }\frac{f(u)}{\sigma (u)}=k< \infty \biggr\} . \end{aligned}$$

Note that \(C_{\sigma }[0,\infty )\) is a normed space with the norm given by \(\|f\|_{\sigma }=\sup_{u\geq 0 }\frac{|f(u)|}{\sigma (u)}\).

Theorem 3.1

Take the sequences of positive numbers \(q=q_{s}\), \(p=p_{s}\) satisfying \(q_{s}\in (0,1)\), \(p_{s}\in (q_{s},1]\) such that \(\lim_{s\rightarrow \infty }q_{s}= 1\), \(\lim_{s\rightarrow \infty }p_{s}= 1\). Then, \(\mathcal{P}_{s,p_{s},q_{s}}^{\tau }\) is uniformly convergent on each compact subset of \(\ [0,\infty )\) and such that

$$ \lim_{s\rightarrow \infty }\mathcal{P}_{s,p_{s},q_{s}}^{\tau }(f;u)=f(u), $$

where \(f\in C[0,\infty )\cap \mathcal{L}\).


To prove the uniform convergence on each compact subset of \([0,\infty )\), it is obvious from the well-known Korovkin’s theorem [23] that \(\lim_{s\rightarrow \infty }\mathcal{P}_{s,p_{s},q_{s}}^{\ell ,\tau } (t^{\eta };u )=u^{\eta }\) for \(\eta =0,1,2\). Whenever, \(q_{s}= 1 \), \(p_{s}=1\) as \(s\rightarrow \infty \), then clearly for all \(i=1,2\) we have \(\frac{1}{[s-i]_{p_{s},q_{s}}}\rightarrow 0\), \(\frac{[s]_{p_{s},q_{s}}}{[s-i]_{p_{s},q_{s}}}\rightarrow 1\), which imply that

$$ \lim_{s\rightarrow \infty }\mathcal{P}_{s,p_{s},q_{s}}^{\tau }(1;u)=1, \qquad \lim_{s\rightarrow \infty }\mathcal{P}_{s,p_{s},q_{s}}^{\tau }(t;u)=u, \qquad \lim_{s\rightarrow \infty }\mathcal{P}_{s,p_{s},q_{s}}^{ \tau } \bigl(t^{2};u\bigr)=u^{2}. $$


Theorem 3.2

For each \(f\in C_{\sigma }^{k}[0,\infty )\), consider the sequences of positive numbers \(0< q_{s}< p_{s}\leq 1\) such that \(\lim_{s\rightarrow \infty }q_{s}=1\), \(\lim_{s\rightarrow \infty }p_{s}=1\). Then the operators \(\mathcal{P}_{s,p_{s},q_{s}}^{\tau }\) satisfy

$$ \lim_{s\rightarrow \infty } \bigl\Vert \mathcal{P}_{s,p_{s},q_{s}}^{\tau }(f)-f \bigr\Vert _{\sigma }=0. $$


We take \(f(t)=t^{\eta }\) with \(\eta =0,1,2\). From Theorem 3.1, since \(\mathcal{P}_{s,p_{s},q_{s}}^{\tau }(t^{\eta };u)\) is uniformly convergent to \(u^{\eta }\) for all \(\eta =0,1,2\), and applying Lemma 2.2, we conclude that

$$ \lim_{s\rightarrow \infty } \bigl\Vert \mathcal{P}_{s,p_{s},q_{s}}^{\tau } ( 1 ) -1 \bigr\Vert _{\sigma }=0. $$

For \(\eta =1\),

$$\begin{aligned} \bigl\Vert \mathcal{P}_{s,p_{s},q_{s}}^{\tau } ( t ) -u \bigr\Vert _{\sigma } & =\sup_{u\geq 0} \frac{ \vert \mathcal{P}_{s,p_{s},q_{s}}^{\tau }(t;u)-u \vert }{1+u^{2}} \\ & \leq \biggl( \frac{{}[ s]_{p_{s},q_{s}}}{[s-1]_{p_{s},q_{s}}}-1 \biggr) \sup _{u\geq 0}\frac{u}{1+u}+ \frac{1}{[s-1]_{p_{s},q_{s}}}\sup_{u\geq 0}\frac{1}{1+u}. \end{aligned}$$


$$ \lim_{s\rightarrow \infty } \bigl\Vert \mathcal{P}_{s,p_{s},q_{s}}^{\tau } ( t ) -u \bigr\Vert _{\sigma }=0. $$

Similarly, if we take \(\eta =2\), then

$$\begin{aligned}& \begin{aligned} \bigl\Vert \mathcal{P}_{s,p_{s},q_{s}}^{\tau } \bigl( t^{2} \bigr) -u^{2} \bigr\Vert _{\sigma } ={}& \sup_{u\geq 0} \frac{ \vert \mathcal{P}_{s,p_{s},q_{s}}^{\tau }(t^{2};u)-u^{2} \vert }{1+u^{2}} \\ \leq {}& \biggl( \frac{[s]_{p_{s},q_{s}}^{2}}{[s-1]_{p_{s},q_{s}}[s-2]_{p,q}}-1 \biggr)\sup _{u\geq 0}\frac{u^{2}}{1+u^{2}} \\ &{}+\frac{[s]_{p_{s},q_{s}}}{[s-1]_{p_{s},q_{s}}[s-2]_{p_{s},q_{s}}} \bigl(1+[2]_{p_{s},q_{s}}+[1+2\tau ]_{p_{s},q_{s}} \bigr)\sup_{u\geq 0} \frac{u}{1+u^{2}} \\ &{}+\frac{[2]_{p_{s},q_{s}}}{[s-1]_{p_{s},q_{s}}[s-2]_{p,q}}\sup_{u \geq 0}\frac{1}{1+u^{2}}, \end{aligned} \\& \lim_{s\rightarrow \infty } \bigl\Vert \mathcal{P}_{s,p_{s},q_{s}}^{\tau } \bigl( t^{2} \bigr) -u^{2} \bigr\Vert _{\sigma }=0. \end{aligned}$$

This completes the proof. □


$$ \omega _{\mu }(f;\delta )=\sup_{ \vert t-u \vert \leq \delta }\sup _{u,t \in {}[ 0,\mu ]} \bigl\vert f(t)-f(u) \bigr\vert . $$

It is obvious that \(\lim_{\delta \rightarrow 0+}\omega _{\mu }(f;\delta )=0\) and for \(f\in C[0,\infty )\),

$$ \bigl\vert f(t)-f(u) \bigr\vert \leq \biggl( \frac{ \vert t-u \vert }{\delta }+1 \biggr) \omega _{\mu }(f;\delta ). $$

Theorem 3.3

Let \(f\in C_{\sigma }[0,\infty )\), and \(0< q_{s}< p_{s}\leq 1 \) be such that \(\lim_{s\rightarrow \infty } q_{s}= 1\), \(\lim_{s\rightarrow \infty }p_{s}= 1\). Moreover, suppose \(\omega _{\mu }(f;\delta )\) is defined by (3.5) on the interval \([0,\mu +1]\subset {}[ 0,\infty )\), for \(\mu >0\). Then for every \(s>2\), we get

$$ \bigl\vert \mathcal{P}_{s,p_{s},q_{s}}^{\tau }(f;u)-f(u) \bigr\vert \leq 2\omega _{\mu +1} \bigl(f;\delta _{s}(u) \bigr)+6 \mathcal{C}_{f}\bigl(1+ \mu ^{2}\bigr) \bigl( \delta _{s}(u) \bigr) ^{2}, $$

where \(\mathcal{C}_{f}\) is a constant depending only on f and \(\delta _{s}(u)=\sqrt{\mathcal{P}_{s,p_{s},q_{s}}^{\tau } (\Phi _{2};u )}\).


For \(u\in {}[ 0,\mu ]\) and \(t\leq \mu +1\), with \(\mu >0\), we have

$$ \bigl\vert f(t)-f(u) \bigr\vert \leq \mathcal{C}_{f} \bigl( 2+u^{2}+t^{2} \bigr) \leq 6\mathcal{C}_{f} \bigl(1+\mu ^{2}\bigr) (t-u)^{2}. $$

Furthermore, for any \(\delta >0\), \(u\in {}[ 0,\mu ]\), and \(t>\mu +1\), with \(\mu >0\),

$$ \bigl\vert f(t)-f(u) \bigr\vert \leq \omega _{\mu +1}\bigl(f; \vert t-u \vert \bigr) \leq \biggl( 1+ \frac{ \vert t-u \vert }{\delta } \biggr)\omega _{ \mu +1}(f;\delta ). $$

From (3.7) and (3.8), we have

$$ \bigl\vert f(t)-f(u) \bigr\vert \leq 6 \mathcal{C}_{f}\bigl(1+\mu ^{2}\bigr) (t-u)^{2}+ \biggl( 1+\frac{ \vert t-u \vert }{\delta } \biggr)\omega _{\mu +1}(f; \delta ). $$

Applying operators \(\mathcal{P}_{s,p_{s},q_{s}}^{\tau }\) and the well-known Cauchy–Schwartz inequality, we have

$$\begin{aligned} \mathcal{P}_{s,p_{s},q_{s}}^{\tau } \bigl( \bigl\vert f(t)-f(u) \bigr\vert ;u \bigr) \leq &6\mathcal{C}_{f}\bigl(1+\mu ^{2}\bigr)\mathcal{P}_{s,p_{s},q_{s}}^{ \tau } ( \Phi _{2};u ) \\ &{}+\mathcal{P}_{s,p_{s},q_{s}}^{\tau } \biggl( 1+ \frac{ \vert t-u \vert }{\delta };u \biggr) \omega _{\mu +1}(f;\delta ). \end{aligned}$$

Moreover, for any \(g\in C_{\sigma }[0,\infty )\), we know

$$\begin{aligned} \mathcal{P}_{s,p_{s},q_{s}}^{\tau }(g;u)-g(u) =&\mathcal{P}_{s,p_{s},q_{s}}^{\tau }(g;u)-g(u) \mathcal{P}_{s,p_{s},q_{s}}^{\tau }(1;u) \\ =&\mathcal{P}_{s,p_{s},q_{s}}^{\tau } \bigl(g(t)-g(u);u \bigr) \\ \leq &\mathcal{P}_{s,p_{s},q_{s}}^{\tau } \bigl( \bigl\vert g(t)-g(u) \bigr\vert ;u \bigr). \end{aligned}$$


$$\begin{aligned} \bigl\vert \mathcal{P}_{s,p_{s},q_{s}}^{\tau }(f;u)-f(u) \bigr\vert \leq &6\mathcal{C}_{f}\bigl(1+\mu ^{2}\bigr)\mathcal{P}_{s,p_{s},q_{s}}^{\tau }(t-u)^{2} \\ &{}+ \biggl(1+\frac{1}{\delta }\mathcal{P}_{s,p_{s},q_{s}}^{\tau } \bigl( \vert t-u \vert ;u \bigr) \biggr)\omega _{\mu +1}(f;\delta ) \\ \leq &6\mathcal{C}_{f}\bigl(1+\mu ^{2}\bigr) \mathcal{P}_{s,p_{s},q_{s}}^{\tau } (\Phi _{2};u ) \\ &{}+ \biggl(1+\frac{1}{\delta }\mathcal{P}_{s,p_{s},q_{s}}^{\tau } ( \Phi _{2};u )^{\frac{1}{2}} \biggr)\omega _{\mu +1}(f; \delta ), \end{aligned}$$


$$ \mathcal{P}_{s,p_{s},q_{s}}^{\tau } \bigl( \vert t-u \vert ;u \bigr)\leq \mathcal{P}_{s,p_{s},q_{s}}^{\tau } (1;u )^{\frac{1}{2}}\mathcal{P}_{s,p_{s},q_{s}}^{\tau } \bigl((t-u)^{2};u \bigr)^{\frac{1}{2}}= \mathcal{P}_{s,p_{s},q_{s}}^{\tau } (\Phi _{2};u )^{\frac{1}{2}}. $$

Finally, if we choose \(\delta =\delta _{s}(u)=\sqrt{\mathcal{P}_{s,p_{s},q_{s}}^{\tau } (\Phi _{2};u )}\), then we get the desired result. □

4 Rate of convergence

In 1963, to measure the smoothness, a mathematical formula of a certain functional was given by Peetre [41]. For all \(\delta >0\) and \(f\in C[0,\infty )\), Peetre defined the K-functional, which we write as \(K_{2}(f;\delta )\). The formulas below give its definition, as well as a bound for some constant \(\mathcal{C}>0\) and the second-order modulus of continuity \(\omega _{2}(f;\delta )\) defined as follows:

$$\begin{aligned}& K_{2}(f;\delta )=\inf_{u\geq 0} \bigl\{ \bigl( \Vert f-\psi \Vert _{C_{B}[0, \infty )}+\delta \bigl\Vert \psi ^{\prime \prime } \bigr\Vert _{C_{B}[0,\infty )} \bigr) :\psi \in C_{B}^{2}[0, \infty ) \bigr\} , \end{aligned}$$
$$\begin{aligned}& K_{2}(f;\delta )\leq \mathcal{C}\bigl\{ \omega _{2}(f; \sqrt{\delta })+\min (1, \delta ) \Vert f \Vert _{C_{B}[0,\infty )}\bigr\} , \\& \omega _{2}(f;\delta )=\sup_{0< v< \delta }\sup _{u\geq 0} \bigl\vert f(u+2v)-2f(u+v)+f(u) \bigr\vert . \end{aligned}$$

Theorem 4.1

Let \(q=q_{s}\), \(p=p_{s}\) with \(q_{s}\in (0,1)\), \(p_{s}\in (q_{s},1] \) and \(\mathcal{R}_{s,p,q}^{\tau }(f;u)=\mathcal{P}_{s,p,q}^{\tau }(f;u)+f(u)-f ( \frac{[s]_{p,q}u+1}{[s-1]_{p,q}} ) \). Then, for every \(\psi \in C_{B}^{2}[0,\infty )\) and \(s>2\), we have

$$ \bigl\vert \mathcal{R}_{s,p_{s},q_{s}}^{\tau }(\psi ;u)-\psi (u) \bigr\vert \leq \chi _{n}(u) \bigl\Vert \psi ^{\prime \prime } \bigr\Vert , $$

where \(\chi _{n}(u)=\delta _{s}^{2}(u)+ ( \mathcal{P}_{s,p,q}(\Phi _{1};u) ) ^{2}\), in which \(\delta _{s}(u)\) is defined in Theorem 3.3and \(\mathcal{P}_{s,p,q}(\Phi _{1};u)\) is defined by Lemma 2.3.


Let \(\psi \in C_{B}^{2}[0,\infty )\). We easily get \(\mathcal{R}_{s,p_{s},q_{s}}^{\tau }(1;u)=1\) and

$$ \mathcal{R}_{s,p_{s},q_{s}}^{\tau }(t;u)=\mathcal{P}_{s,p_{s},q_{s}}^{ \tau }(t;u)+u- \biggl(\frac{[s]_{p_{s},q_{s}}u+1}{[s-1]_{p_{s},q_{s}}} \biggr)=u. $$


$$\begin{aligned}& \bigl\Vert \mathcal{P}_{s,p_{s},q_{s}}^{\tau }(f;u) \bigr\Vert \leq \Vert f \Vert , \\& \bigl\vert \mathcal{R}_{s,p_{s},q_{s}}^{\tau }(f;u) \bigr\vert \leq \bigl\vert \mathcal{P}_{s,p_{s},q_{s}}^{\tau }(f;u) \bigr\vert + \bigl\vert f(u) \bigr\vert - \biggl\vert f \biggl( \frac{[s]_{p_{s},q_{s}}u+1}{[s-1]_{p_{s},q_{s}}} \biggr) \biggr\vert \leq 3 \Vert f \Vert . \end{aligned}$$

From the Taylor series expansion, we have

$$ \psi (t)=\psi (u)+(t-u)\psi ^{\prime }(u)+ \int _{u}^{t}(t-\alpha ) \psi ^{\prime \prime }(\alpha )\,\mathrm{d}\alpha . $$

Applying the operator \(\mathcal{R}_{s,p_{s},q_{s}}^{\tau }\), we conclude that

$$\begin{aligned}& \begin{aligned} \mathcal{R}_{s,p_{s},q_{s}}^{\tau }(\psi ;u)-\psi (u)={}&\psi ^{ \prime }(u)\mathcal{R}_{s,p_{s},q_{s}}^{\tau }(t-u;u)+ \mathcal{R}_{s,p_{s},q_{s}}^{ \tau } \biggl( \int _{u}^{t}(t-\alpha )\psi ^{\prime \prime }( \alpha )\, \mathrm{d}\alpha ;u \biggr) \\ ={}&\mathcal{R}_{s,p_{s},q_{s}}^{\tau } \biggl( \int _{u}^{t}(t-\alpha ) \psi ^{\prime \prime }(\alpha )\,\mathrm{d}\alpha ;u \biggr) \\ ={}&\mathcal{P}_{s,p_{s},q_{s}}^{\tau } \biggl( \int _{u}^{t}(t-\alpha ) \psi ^{\prime \prime }(\alpha )\,\mathrm{d}\alpha ;u \biggr) \\ &{}- \int _{u}^{\frac{[s]_{p_{s},q_{s}}u+1}{[s-1]_{p_{s},q_{s}}}} \biggl( \frac{[s]_{p_{s},q_{s}}u+1}{[s-1]_{p_{s},q_{s}}}- \alpha \biggr)\psi ^{ \prime \prime }(\alpha )\,\mathrm{d}\alpha \end{aligned}\\& \begin{aligned} \bigl\vert \mathcal{R}_{s,p_{s},q_{s}}^{\tau }(\psi ;u)-\psi (u) \bigr\vert \leq{} & \biggl\vert \mathcal{P}_{s,p_{s},q_{s}}^{\tau } \biggl( \int _{u}^{t}(t- \alpha )\psi ^{\prime \prime }(\alpha )\,\mathrm{d}\alpha ;u \biggr) \biggr\vert \\ &{}+ \biggl\vert \int _{u}^{ \frac{[s]_{p_{s},q_{s}}u+1}{[s-1]_{p_{s},q_{s}}}} \biggl(\frac{[s]_{p_{s},q_{s}}u+1}{[s-1]_{p_{s},q_{s}}}-\alpha \biggr)\psi ^{ \prime \prime }(\alpha )\,\mathrm{d} \alpha \biggr\vert . \end{aligned} \end{aligned}$$


$$ \biggl\vert \int _{u}^{t}(t-\alpha )\psi ^{\prime \prime }( \alpha ) \,\mathrm{d}\alpha \biggr\vert \leq (t-u)^{2} \bigl\Vert \psi ^{\prime \prime } \bigr\Vert , $$

we conclude that

$$ \biggl\vert \int _{u}^{\frac{[s]_{p_{s},q_{s}}u+1}{[s-1]_{p_{s},q_{s}}}} \biggl(\frac{[s]_{p_{s},q_{s}}u+1}{[s-1]_{p_{s},q_{s}}}-\alpha \biggr)\psi ^{ \prime \prime }(\alpha )\,\mathrm{d} \alpha \biggr\vert \leq \biggl( \frac{[s]_{p_{s},q_{s}}u+1}{[s-1]_{p_{s},q_{s}}}-u \biggr)^{2} \bigl\Vert \psi ^{ \prime \prime } \bigr\Vert . $$


$$ \bigl\vert \mathcal{R}_{s,p_{s},q_{s}}^{\tau }(\psi ;u)-\psi (u) \bigr\vert \leq {\biggl\{ }\mathcal{P}_{s,p_{s},q_{s}}^{\tau } \bigl( (t-u)^{2};u \bigr) + \biggl(\frac{[s]_{p_{s},q_{s}}u+1}{[s-1]_{p_{s},q_{s}}}-u \biggr)^{2} {\biggr\} } \bigl\Vert \psi ^{\prime \prime } \bigr\Vert . $$

Thus we complete the proof. □

Theorem 4.2

Let \(q=q_{s}\), \(p=p_{s}\) with \(q_{s}\in (0,1)\), \(p_{s}\in (q_{s},1]\) and \(f\in C_{B}[0,\infty )\). Then, for every \(\psi \in C_{B}^{2}[0,\infty )\) and \(s>2\) there exits a positive constant \(\mathcal{C}> \) satisfying the inequality

$$\begin{aligned} \bigl\vert \mathcal{P}_{s,p_{s},q_{s}}^{\tau }(f ;u)-f(u) \bigr\vert \leq &\mathcal{A} \biggl\{ \omega _{2} \biggl( f; \frac{\sqrt{\chi _{s}(u)}}{2} \biggr) +\min \biggl( 1, \frac{\chi _{s}(u)}{4} \biggr) \Vert f \Vert \biggr\} \\ &{}+\omega \bigl(f; \bigl\vert \mathcal{P}_{s,p_{s},q_{s}}(\Phi _{1};u) \bigr\vert \bigr). \end{aligned}$$


For all \(f\in C_{B}[0,\infty )\) and \(\psi \in C_{B}^{2}[0,\infty )\), it is very easy to see the result from Theorem 4.1. Indeed,

$$\begin{aligned} \bigl\vert \mathcal{R}_{s,p_{s},q_{s}}^{\tau }(f ;u)-f(u) \bigr\vert =& \biggl\vert \mathcal{R}_{s,p_{s},q_{s}}^{\tau }(f ;u)-f(u)+f \biggl( \frac{[s]_{p_{s},q_{s}}u+1}{[s-1]_{p,q}} \biggr)-f(u) \biggr\vert \\ \leq & \bigl\vert \mathcal{R}_{s,p_{s},q_{s}}^{\tau }(f-\psi ;u) \bigr\vert + \bigl\vert \mathcal{R}_{s,p_{s},q_{s}}^{\tau }( \psi ;u)-\psi (u) \bigr\vert \\ &{}+ \bigl\vert \psi (u)-f(u) \bigr\vert + \biggl\vert f \biggl( \frac{[s]_{p_{s},q_{s}}u+1}{[s-1]_{p_{s},q_{s}}} \biggr)-f(u) \biggr\vert \\ \leq &4 \Vert f-\psi \Vert +\chi _{s}(u) \bigl\Vert \psi ^{ \prime \prime } \bigr\Vert \\ &{}+\omega \biggl(f; \biggl\vert \biggl( \frac{[s]_{p_{s},q_{s}}}{[s-1]_{p_{s},q_{s}}}-1 \biggr) u+ \frac{1}{[s-1]_{p_{s},q_{s}}} \biggr\vert \biggr). \end{aligned}$$

By taking the infimum over all \(\psi \in C_{B}^{2}[0,\infty )\) and using (4.1), we get

$$\begin{aligned} \bigl\vert \mathcal{R}_{s,p_{s},q_{s}}^{\tau }(f ;u)-f(u) \bigr\vert \leq &4K_{2} \biggl( f;\frac{\chi _{s}(u)}{4} \biggr) +\omega \biggl(f; \biggl\vert \biggl( \frac{[s]_{p_{s},q_{s}}}{[s-1]_{p_{s},q_{s}}}-1 \biggr) u+ \frac{1}{[s-1]_{p_{s},q_{s}}} \biggr\vert \biggr) \\ \leq &\mathcal{A} {\biggl\{ }\omega _{2} \biggl( f; \frac{\sqrt{\chi _{s}(u)}}{2} \biggr) +\min \biggl( 1;\frac{\chi _{s}(u)}{4} \biggr) \Vert f \Vert {\biggr\} } \\ &{}+\omega \biggl(f; \biggl\vert \biggl( \frac{[s]_{p_{s},q_{s}}}{[s-1]_{p_{s},q_{s}}}-1 \biggr) u+ \frac{1}{[s-1]_{p_{s},q_{s}}} \biggr\vert \biggr). \end{aligned}$$


We consider the following Lipschitz-type maximal function [24] and obtain the local approximation. For \(f\in C[0,\infty ]\), \(0<\kappa \leq 1\) and \(t,u\geq 0\), we recall that

$$ \mathrm{Lip}_{M}(\kappa )=\bigl\{ f: \bigl\vert f(t)-f(u) \bigr\vert \leq M \vert t-u \vert ^{ \kappa } \bigr\} . $$

Theorem 4.3

For all \(\kappa \in (0,1]\), \(s>2\), and \(f\in C_{B}[0,\infty )\), we have

$$ \bigl\vert \mathcal{P}_{s,p_{s},q_{s}}^{\tau }(f;u)-f(u) \bigr\vert \leq M \bigl( \delta _{s}(u) \bigr) ^{\kappa }, $$

where \(\delta _{s}(u)\) is given in Theorem 3.3.


We prove the claim by applying (4.4) and the well-known Hölder’s inequality:

$$\begin{aligned} \bigl\vert \mathcal{P}_{s,p_{s},q_{s}}^{\tau }(f;u)-f(u) \bigr\vert \leq &\mathcal{P}_{s,p_{s},q_{s}}^{\tau } \bigl( \bigl\vert f(t)-f(u) \bigr\vert ;u \bigr) \\ \leq &M| \mathcal{P}_{s,p_{s},q_{s}}^{\tau } \bigl( \vert t-u \vert ^{\kappa };u \bigr) \\ \leq &M \bigl( \mathcal{P}_{s,p_{s},q_{s}}^{\tau }(1;u) \bigr) ^{ \frac{2-\kappa }{2}} \bigl( \mathcal{P}_{s,p_{s},q_{s}}^{\tau }\bigl( \vert t-u \vert ^{2};u\bigr) \bigr) ^{\frac{\kappa }{2}} \\ =&M \bigl( \mathcal{P}_{s,p_{s},q_{s}}^{\tau }(\Phi _{2};u) \bigr) ^{ \frac{\kappa }{2}}. \end{aligned}$$

This gives the desired result. □

We denote

$$\begin{aligned}& C_{B}^{2}[0,\infty )= \bigl\{ \psi :\psi \in C_{B}[0,\infty ) \text{ and } \psi^{\prime },\psi ^{\prime \prime }\in C_{B}[0,\infty ) \bigr\} , \end{aligned}$$
$$\begin{aligned}& \Vert \psi \Vert _{C_{B}^{2}[0,\infty )}= \Vert \psi \Vert _{C_{B}[0,\infty )}+ \bigl\Vert \psi ^{\prime } \bigr\Vert _{C_{B}[0, \infty )}+ \bigl\Vert \psi ^{\prime \prime } \bigr\Vert _{C_{B}[0,\infty )}, \end{aligned}$$
$$\begin{aligned}& \Vert \psi \Vert _{C_{B}[0,\infty )}=\sup_{u\geq 0} \bigl\vert \psi (u) \bigr\vert . \end{aligned}$$

Theorem 4.4

Let the positive sequences of numbers \(0< q_{s}< p_{s}\leq 1\) satisfy \(\lim_{s\rightarrow \infty }q_{s}= 1\), \(\lim_{s\rightarrow \infty }p_{s}= 1\). Then for all \(\psi \in C_{B}^{2}[0,\infty )\), the operators \(\mathcal{P}_{s,p_{s},q_{s}}^{\tau }\) have the property

$$ \bigl\vert \mathcal{P}_{s,p_{s},q_{s}}^{\tau }(\psi ;u)-\psi (u) \bigr\vert \leq \Theta _{s}(u) \Vert \psi \Vert _{C_{B}^{2}[0,\infty )}, $$

where \(\Theta _{s}(u)=\sqrt{\delta _{s}(u)}+ \frac{ (\delta _{s}(u) )^{2}}{2}\).


Let \(\psi \in C_{B}^{2}[0,\infty )\). Then

$$ \psi (t)=\psi (u)+\psi ^{\prime }(u) (t-u)+\psi ^{\prime \prime }( \varphi ) \frac{(t-u)^{2}}{2}\quad \text{for } \varphi \in (u,t), $$

where if we take

$$\begin{aligned}& \mathcal{S}=\sup_{u\geq 0} \bigl\vert \psi ^{\prime }(u) \bigr\vert = \bigl\Vert \psi ^{\prime } \bigr\Vert _{C_{B}[0,\infty )}\leq \Vert \psi \Vert _{C_{B}^{2}[0, \infty )}, \\& \mathcal{T}=\sup_{u\geq 0} \bigl\vert \psi ^{\prime \prime }(u) \bigr\vert = \bigl\Vert \psi ^{\prime \prime } \bigr\Vert _{C_{B}[0,\infty )} \leq \Vert \psi \Vert _{C_{B}^{2}[0, \infty )}, \end{aligned}$$

then we have

$$\begin{aligned} \bigl\vert \psi (t)-\psi (u) \bigr\vert \leq & \mathcal{S} \vert t-u \vert +\frac{1}{2}\mathcal{T}(t-u)^{2} \\ \leq & \biggl( \vert t-u \vert +\frac{1}{2}(t-u)^{2} \biggr) \Vert \psi \Vert _{C_{B}^{2}[0,\infty )}. \end{aligned}$$


$$\begin{aligned} \bigl\vert \mathcal{P}_{s,p_{s},q_{s}}^{\tau }(\psi ;u)-\psi (u) \bigr\vert \leq & \biggl( \mathcal{P}_{s,p_{s},q_{s}}^{\tau }\bigl( \vert t-u \vert ;u\bigr)+\frac{1}{2}\mathcal{P}_{s,p_{s},q_{s}}^{\tau } \bigl((t-u)^{2};u \bigr) \biggr) \Vert \psi \Vert _{C_{B}^{2}[0,\infty )} \\ \leq & \biggl( \bigl(\mathcal{P}_{s,p_{s},q_{s}}^{\tau } ( \Phi _{2};u ) \bigr)^{\frac{1}{2}}+\frac{1}{2} \mathcal{P}_{s,p_{s},q_{s}}^{\tau } (\Phi _{2};u ) \biggr) \Vert \psi \Vert _{C_{B}^{2}[0,\infty )}. \end{aligned}$$

This completes the proof of Theorem 4.4. □

5 Conclusion

We constructed a \((p,q)\)-variant of Szász operators by using the Beta functions of the second kind by introducing the Dunkl generalization. We obtained the approximation results involving local and global approximations in Korovkin’s and weighted Korovkin’s spaces. We applied some techniques of earlier investigation and discussed the convergence of operators by employing the modulus of continuity, Lipschitz class and Peetre’s K-functionals.

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  1. Acar, T.: \((p,q)\)-Generalization of Szász–Mirakyan operators. Math. Methods Appl. Sci. 39(10), 2685–2695 (2016)

    Article  MathSciNet  Google Scholar 

  2. Acar, T., Agrawal, P.N., Kumar, A.S.: On a modification of \((p,q)\)-Szász–Mirakyan operators. Complex Anal. Oper. Theory 12(1), 155–167 (2018)

    Article  MathSciNet  Google Scholar 

  3. Acar, T., Aral, A., Mohiuddine, S.A.: On Kantorovich modification of \((p,q)\)-Baskakov operators. J. Inequal. Appl. 2016, 98 (2016)

    Article  MathSciNet  Google Scholar 

  4. Acar, T., Aral, A., Mohiuddine, S.A.: On Kantorovich modification of \((p,q)\)-Bernstein operators. Iran. J. Sci. Technol., Trans. A, Sci. 42, 1459–1464 (2018)

    Article  MathSciNet  Google Scholar 

  5. Acar, T., Aral, A., Mohiuddine, S.A.: Approximation by bivariate \((p,q)\)-Bernstein–Kantorovich operators. Iran. J. Sci. Technol. Trans. A, Sci. 42(2), 655–662 (2018)

    Article  MathSciNet  Google Scholar 

  6. Acar, T., Aral, A., Mursaleen, M.: Approximation by Baskakov–Durrmeyer operators based on \((p,q)\)-integers. Math. Slovaca 68(4), 897–906 (2018)

    Article  MathSciNet  Google Scholar 

  7. Acar, T., Aral, A., Raşa, I.: Positive linear operators preserving τ and \(\tau ^{2}\). Constr. Math. Anal. 2(3), 98–102 (2019)

    MathSciNet  Google Scholar 

  8. Acar, T., Mohiuddine, S.A., Mursaleen, M.: Approximation by \((p,q) \)-Baskakov–Durrmeyer–Stancu operators. Complex Anal. Oper. Theory 12(6), 1453–1468 (2018)

    Article  MathSciNet  Google Scholar 

  9. Acar, T., Mursaleen, M., Mohiuddine, S.A.: Stancu type \((p,q)\)-Szász–Mirakyan–Baskakov operators. Commun. Fac. Sci. Univ. Ank. Sér. A1 Math. Stat. 67(1), 116–128 (2018)

    MathSciNet  Google Scholar 

  10. Alotaibi, A., Nasiruzzaman, M., Mursaleen, M.: A Dunkl type generalization of Szász operators via post-quantum calculus. J. Inequal. Appl. 2018, 287 (2018)

    Article  Google Scholar 

  11. Bin Jebreen, H., Mursaleen, M., Naaz, A.: Approximation by quaternion \((p,q)\)-Bernstein polynomials and Voronovskaja type result on compact disk. Adv. Differ. Equ. 2018, 448 (2018)

    Article  MathSciNet  Google Scholar 

  12. De Sole, A., Kac, V.G.: On integral representations of q-gamma and q-beta functions. Atti Accad. Naz. Lincei, Rend. Lincei, Mat. Appl. 16, 11–29 (2005)

    MathSciNet  Google Scholar 

  13. İçöz, G., Çekim, B.: Dunkl generalization of Szász operators via q-calculus. J. Inequal. Appl. 2015, 284 (2015)

    Article  Google Scholar 

  14. İnce İlarslan, H.G., Acar, T.: Approximation by bivariate \((p,q)\)-Baskakov–Kantorovich operators. Georgian Math. J. 25(3), 397–407 (2018)

    Article  MathSciNet  Google Scholar 

  15. Kadak, U.: On weighted statistical convergence based on \((p,q)\)-integers and related approximation theorems for functions of two variables. J. Math. Anal. Appl. 443(2), 752–764 (2016)

    Article  MathSciNet  Google Scholar 

  16. Kadak, U.: Weighted statistical convergence based on generalized difference operator involving \((p,q)\)-gamma function and its applications to approximation theorems. J. Math. Anal. Appl. 448(2), 1633–1650 (2017)

    Article  MathSciNet  Google Scholar 

  17. Kadak, U., Mishra, V.N., Pandey, S.: Chlodowsky type generalization of \((p,q)\)-Szász operators involving Brenke type polynomials. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 112(4), 1443–1462 (2018)

    Article  MathSciNet  Google Scholar 

  18. Kadak, U., Mohiuddine, S.A.: Generalized statistically almost convergence based on the difference operator which includes the \((p,q) \)-Gamma function and related approximation theorems. Results Math. 73(9) (2018).

  19. Khan, A., Sharma, V.: Statistical approximation by \((p,q)\)-analogue of Bernstein–Stancu operators. Azerb. J. Math. 8(2), 100–121 (2018)

    MathSciNet  Google Scholar 

  20. Khan, K., Lobiyal, D.K.: Bézier curves based on Lupaş \((p,q)\)-analogue of Bernstein functions in CAGD. J. Comput. Appl. Math. 317, 458–477 (2017)

    Article  MathSciNet  Google Scholar 

  21. Khan, K., Lobiyal, D.K., Kilicman, A.: Bézier curves and surfaces based on modified Bernstein polynomials. Azerb. J. Math. 9(1), 3–21 (2019)

    MathSciNet  Google Scholar 

  22. Kilicman, A., Ayman Mursaleen, M., Al-Abied, A.H.H.: Stancu type Baskakov–Durrmeyer operators and approximation properties. Mathematics 8, Article ID 1164 (2020).

    Article  Google Scholar 

  23. Korovkin, P.P.: On convergence of linear operators in the space of continuous functions. Dokl. Akad. Nauk SSSR 90, 961–964 (1953) (Russian)

    MathSciNet  Google Scholar 

  24. Lenze, B.: On Lipschitz type maximal functions and their smoothness spaces. Nederl. Akad. Indag. Math. 50, 53–63 (1988)

    Article  MathSciNet  Google Scholar 

  25. Lupaş, A.: A q-analogue of the Bernstein operator. Univ. Cluj-Napoca Seminar Numer. Stat., Calculus 9, 85–92 (1987)

    MathSciNet  Google Scholar 

  26. Maurya, R., Sharma, H., Gupta, C.: Approximation properties of Kantorovich type modifications of \((p,q)\)-Meyer–König–Zeller operators. Constr. Math. Anal. 1(1), 58–72 (2018)

    MathSciNet  Google Scholar 

  27. Milovanovic, G.V., Mursaleen, M., Nasiruzzaman, M.: Modified Stancu type Dunkl generalization of Szász–Kantorovich operators. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 112, 135–151 (2018)

    Article  MathSciNet  Google Scholar 

  28. Mohiuddine, S.A., Acar, T., Alghamdi, M.A.: Genuine modified Bernstein–Durrmeyer operators. J. Inequal. Appl. 2018, 104 (2018)

    Article  MathSciNet  Google Scholar 

  29. Mohiuddine, S.A., Acar, T., Alotaibi, A.: Construction of a new family of Bernstein–Kantorovich operators. Math. Methods Appl. Sci. 40, 7749–7759 (2017)

    Article  MathSciNet  Google Scholar 

  30. Mohiuddine, S.A., Acar, T., Alotaibi, A.: Durrmeyer type \((p,q)\)-Baskakov operators preserving linear functions. J. Math. Inequal. 12, 961–973 (2018)

    Article  MathSciNet  Google Scholar 

  31. Mohiuddine, S.A., Alamri, B.A.S.: Generalization of equi-statistical convergence via weighted lacunary sequence with associated Korovkin and Voronovskaya type approximation theorems. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 113(3), 1955–1973 (2019)

    Article  MathSciNet  Google Scholar 

  32. Mohiuddine, S.A., Ōzger, F.: Approximation of functions by Stancu variant of Bernstein–Kantorovich operators based on shape parameter α. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 114, 70 (2020)

    Article  MathSciNet  Google Scholar 

  33. Mursaleen, M., Ansari, K.J., Khan, A.: On \((p,q)\)-analogue of Bernstein operators. Appl. Math. Comput. 266, 874–882 (2015)

    MathSciNet  Google Scholar 

  34. Mursaleen, M., Ansari, K.J., Khan, A.: Some approximation results by \((p,q)\)-analogue of Bernstein–Stancu operators. Appl. Math. Comput. 264, 392–402 (2015)

    MathSciNet  Google Scholar 

  35. Mursaleen, M., Naaz, A., Khan, A.: Improved approximation and error estimations by King type \((p,q)\)-Szász–Mirakjan–Kantorovich operators. Appl. Math. Comput. 348, 175–185 (2019)

    MathSciNet  Google Scholar 

  36. Mursaleen, M., Nasiruzzaman, M.: Approximation of modified Jakimovski–Leviatan–beta type operators. Constr. Math. Anal. 1(2), 88–98 (2018)

    MathSciNet  Google Scholar 

  37. Mursaleen, M., Nasiruzzaman, Md., Alotaibi, A.: On modified Dunkl generalization of Szász operators via q-calculus. J. Inequal. Appl. 2017, 38 (2017)

    Article  Google Scholar 

  38. Nasiruzzaman, M., Mukheimer, A., Mursaleen, M.: A Dunkl-type generalization of Szász–Kantorovich operators via post quantum calculus. Symmetry 11(2), 232 (2019)

    Article  Google Scholar 

  39. Nasiruzzaman, M., Mukheimer, A., Mursaleen, M.: Approximation results on Dunkl generalization of Phillips operators via q-calculus. Adv. Differ. Equ. 2019, 244 (2019)

    Article  MathSciNet  Google Scholar 

  40. Ōzger, F., Srivastava, H.M., Mohiuddine, S.A.: Approximation of functions by a new class of generalized Bernstein–Schurer operators. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 114, 173 (2020)

    Article  MathSciNet  Google Scholar 

  41. Peetre, J.: A theory of interpolation of normed spaces. Noteas de Mathematica, Instituto de Mathemática Pura e Applicada, Conselho Nacional de Pesquidas, Rio de Janeiro 39 (1968)

  42. Phillips, G.M.: Bernstein polynomials based on the q-integers, The heritage of P.L. Chebyshev, A Festschrift in honor of the 70th-birthday of Professor T.J. Rivlin. Ann. Numer. Math. 4, 511–518 (1997)

    Google Scholar 

  43. Rao, N., Wafi, A.: \((p,q)\)-Bivariate–Bernstein–Chlodowsky operators. Filomat 32(2), 369–378 (2018)

    Article  MathSciNet  Google Scholar 

  44. Rao, N., Wafi, A.: Bivariate–Schurer–Stancu operators based on \((p,q)\)-integers. Filomat 32(4), 1251–1258 (2018)

    Article  MathSciNet  Google Scholar 

  45. Rao, N., Wafi, A., Acu, A.M.: q-Szász–Durrmeyer type operators based on Dunkl analogue. Complex Anal. Oper. Theory 13(3), 915–934 (2019)

    Article  MathSciNet  Google Scholar 

  46. Sucu, S.: Dunkl analogue of Szász operators. Appl. Math. Comput. 244, 42–48 (2014)

    MathSciNet  Google Scholar 

  47. Szász, O.: Generalization of S. Bernstein’s polynomials to the infinite interval. J. Res. Natl. Bur. Stand. 45, 239–245 (1950)

    Article  MathSciNet  Google Scholar 

  48. Wafi, A., Rao Deepmala, N.: Approximation properties of \((p,q)\)-variant of Stancu–Schurer operators. Bol. Soc. Parana. Mat. 37(4), 137–150 (2019)

    Article  MathSciNet  Google Scholar 

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Nasiruzzaman, M., Alotaibi, A. & Mursaleen, M. Dunkl-type generalization of the second kind beta operators via \((p,q)\)-calculus. J Inequal Appl 2021, 6 (2021).

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