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, 37‐39, 45, 46]. Let \([s]_{p,q}\) be the \((p,q)\)-integer defined as
The \((p,q)\)-power basis is explained as
Furthermore, the \((p,q)\)-analogues of the exponential function are defined by
Moreover, the \((p,q)\)-Dunkl analogue of the exponential function is defined by
And a recursion identity is defined as
where
$$\begin{aligned}& [ s ] _{p,q}=p^{s-1}+qp^{s-3}+\cdots +q^{s-1}= \textstyle\begin{cases} \frac{p^{s}-q^{s}}{p-q} & (p\neq q\neq 1), \\ \frac{1-q^{s}}{1-q} & (p=1), \\ s & (p=q=1),\end{cases}\displaystyle \\& (au+bv)_{p,q}^{s}:=\sum_{\ell =0}^{s}p^{ \frac{(s-\ell )(s-\ell -1)}{2}}q^{\frac{\ell (\ell -1)}{2}} \begin{bmatrix} s \\ \ell \end{bmatrix} _{p,q}a^{s-\ell }b^{\ell }u^{s-\ell }v^{\ell }, \\& (1-u)_{p,q}^{s}=(1-u) (p-qu) \bigl(p^{2}-q^{2}u \bigr)\cdots \bigl(p^{s-1}-q^{s-1}u\bigr), \\& (u-y)_{p,q}^{s}=\textstyle\begin{cases} \prod_{j=0}^{s-1}(p^{j}u-q^{j}y) & \text{if }s\in \mathbb{N}, \\ 1 & \text{if }s=0.\end{cases}\displaystyle \end{aligned}$$
(1.1)
$$ (u\oplus v)_{p,q}^{s}=(u+v) (pu+qv) \bigl(p^{2}u+q^{2}v \bigr)\cdots \bigl(p^{s-1}u+q^{s-1}v\bigr). $$
$$ e_{p,q}(u)=\sum_{\ell =0}^{\infty }p^{\frac{\ell (\ell -1)}{2}} \frac{u^{\ell }}{[\ell ]_{p,q}!},\qquad E_{p,q}(u)=\sum _{\ell =0}^{ \infty }q^{\frac{\ell (\ell -1)}{2}}\frac{u^{\ell }}{[\ell ]_{p,q}!}. $$
$$\begin{aligned}& e_{\tau ,p,q}(u)=\sum_{\ell =0}^{\infty }p^{\frac{\ell (\ell -1)}{2}} \frac{u^{\ell }}{\gamma _{\tau ,p,q}(\ell )} , \end{aligned}$$
(1.2)
$$\begin{aligned}& \gamma _{\tau ,p,q}(\ell ) \\& \quad = \frac{\prod_{i=0}^{[\frac{\ell +1}{2}]-1}p^{2\tau (-1)^{i+1}+1} ( (p^{2})^{i}p^{2\tau +1}-(q^{2})^{i}q^{2\tau +1} ) \prod_{j=0}^{[\frac{\ell }{2}]-1}p^{2\tau (-1)^{j}+1} ( (p^{2})^{j}p^{2}-(q^{2})^{j}q^{2} ) }{(p-q)^{\ell }}. \end{aligned}$$
(1.3)
$$ \gamma _{\tau ,p,q}(\ell +1)= \frac{p^{2\tau (-1)^{\ell +1}+1}({p^{2\tau \theta _{\ell +1}+\ell +1}-q^{2\tau \theta _{\ell +1}+\ell +1}})}{(p-q)}\gamma _{\tau ,p,q}(\ell ), $$
(1.4)
$$ \theta _{\ell }= \textstyle\begin{cases} 0 & \text{for }\ell =2m, m=0,1,2,\dots , \\ 1 & \text{for }\ell =2m+1, m=0,1,2,\dots \end{cases} $$
(1.5)
For \(m=0,1,2,\dots s\), the number \([\frac{m}{2}]\) denotes the greatest integer function evaluated at \(m/2\).
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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
$$ D_{s,p,q}(f;u)=\frac{1}{e_{\tau ,p,q}([s]_{p,q}u)}\sum_{\ell =0}^{ \infty }\frac{([s]_{p,q}u)^{\ell }}{\gamma _{\tau ,p,q}(\ell )}p^{ \frac{\ell (\ell -1)}{2}}f \biggl( \frac{p^{\ell +2\tau \theta _{\ell }}-q^{\ell +2\tau \theta _{\ell }}}{p^{\ell -1}(p^{s}-q^{s})} \biggr) . $$
(1.6)
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
where
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
where a formula for the \((p,q)\)-Beta function is given by
$$ \mathcal{P}_{s,p,q}^{\tau }(f;u)=\sum _{\ell =0}^{\infty } \mathcal{Q}_{s,p,q}(u) \frac{1}{\mathcal{B}_{p,q}(\ell +2\tau \theta _{\ell }+1,s)}\int _{0}^{\infty } \frac{t^{\ell +2\tau \theta _{\ell }}}{(1\oplus pt)_{p,q}^{\ell +2\tau \theta _{\ell }+s+1}}f(t) \,\mathrm{d}_{p,q}t, $$
(2.1)
$$ \mathcal{Q}_{s,p,q}(u)=\frac{1}{e_{\tau ,p,q}([s]_{p,q}u)} \frac{([s]_{p,q}u)^{\ell }}{\gamma _{\tau ,p,q}(\ell )}p^{ \frac{\ell (\ell -1)}{2}}, $$
$$ \mathcal{B}_{p,q}(\alpha ,\beta )= \int _{0}^{\infty } \frac{t^{\alpha -1}}{(1\oplus pt)_{p,q}^{\alpha +\beta }} \,\mathrm{d}_{p,q}t,\quad \alpha , \beta \in \mathbb{N,} $$
(2.2)
$$ \mathcal{B}_{p,q}(\alpha ,\beta ) = \frac{[\alpha -1]_{p,q}}{p^{\alpha -1} [\beta ]_{p,q}} \mathcal{B}_{p,q}(\alpha -1,\beta +1), \quad \alpha , \beta \in \mathbb{N}. $$
(2.3)
Moreover, to obtain the basic estimates here, we use the following relations:
$$\begin{aligned}& {}[ \ell +1+2\tau \theta _{\ell }]_{p,q}=q[ \ell +2\tau \theta _{ \ell }]_{p,q}+p^{\ell +2\tau \theta _{\ell }}, \end{aligned}$$
(2.4)
$$\begin{aligned}& {}[ \ell +2+2\tau \theta _{\ell }]_{p,q}=q^{2}[ \ell +2\tau \theta _{ \ell }]_{p,q}+(p+q)p^{\ell +2\tau \theta _{\ell }}. \end{aligned}$$
(2.5)
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For more related results on \((p,q)\)-analogues, we refer to [1‐6, 8, 9, 11, 14‐21, 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:
and
$$ \mathcal{P}_{s,p,q}^{\tau }(f;u)\leq \textstyle\begin{cases} \frac{{}[ s]_{p,q}}{[s-1]_{p,q}}u+\frac{1}{[s-1]_{p,q}} \quad \textit{for } f(t)=t, \\ \frac{{}[ s]_{p,q}^{2}}{[s-1]_{p,q}[s-2]_{p,q}}u^{2}+ \frac{[s]_{p,q}}{[s-1]_{p,q}[s-2]_{p,q}} ( 1+[2]_{p,q}+[1+2\tau ]_{p,q} ) u \\ \quad {}+\frac{[2]_{p,q}}{[s-1]_{p,q}[s-2]_{p,q}} \quad \textit{for } f(t)=t^{2},\end{cases} $$
(2.6)
$$ \mathcal{P}_{s,p,q}^{\tau }(f;u)\geq \textstyle\begin{cases} \frac{q[s]_{p,q}}{[s-1]_{p,q}}u+\frac{1}{[s-1]_{p,q}} \quad \textit{for } f(t)=t, \\ \frac{q^{3}[s]_{p,q}^{2}}{[s-1]_{p,q}[s-2]_{p,q}}u^{2} & \\ \quad {}+\frac{q[s]_{p,q}}{[s-1]_{p,q}[s-2]_{p,q}} (q+[2]_{p,q}+q^{2+2 \tau }[1-2\tau ]_{p,q} \frac{e_{\tau ,p,q} ( \frac{q}{p}[s]_{p,q}u ) }{e_{\tau ,p,q}([s]_{p,q}u)} )u & \\ \quad {}+\frac{[2]_{p,q}}{[s-1]_{p,q}[s-2]_{p,q}} \quad \textit{for } f(t)=t^{2}.\end{cases} $$
Proof
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
Clearly, we have
and
$$\begin{aligned} \mathcal{P}_{s,p,q}^{\tau }(t;u) =&\sum _{\ell =0}^{\infty } \mathcal{Q}_{s,p,q}(u) \frac{1}{\mathcal{B}_{p,q}(\ell +2\tau \theta _{\ell }+1,s)}\int _{0}^{\infty } \frac{t^{\ell +2\tau \theta _{\ell }+1}}{(1\oplus pt)_{p,q}^{\ell +2\tau \theta _{\ell }+s+1}} \,\mathrm{d}_{p,q}t \\ =&\sum_{\ell =0}^{\infty }\mathcal{Q}_{s,p,q}(u) \frac{\mathcal{B}_{p,q}(\ell +2\tau \theta _{\ell }+2,s-1)}{\mathcal{B}_{p,q}(\ell +2\tau \theta _{\ell }+1,s)} \\ =&\frac{q}{[s-1]_{p,q}}\sum_{\ell =0}^{\infty } \mathcal{Q}_{s,p,q}(u) \frac{1}{p^{\ell +2\tau \theta _{\ell }+1}}[\ell +2\tau \theta _{\ell }]_{p,q}+\frac{1}{p[s-1]_{p,q}} \\ =&\frac{1}{p[s-1]_{p,q}}+\frac{q[s]_{p,q}}{p^{2}[s-1]_{p,q}}\sum_{ \ell =0}^{\infty } \mathcal{Q}_{s,p,q}(u) \biggl( \frac{p^{2\ell +2\tau \theta _{2\ell }}-q^{2\ell +2\tau \theta _{2\ell }}}{p^{2\ell -1}(p^{s}-q^{s})} \biggr) \\ &{}+\frac{q[s]_{p,q}}{p^{2+2\tau }[s-1]_{p,q}}\sum_{\ell =0}^{\infty }\mathcal{Q}_{s,p,q}(u) \biggl( \frac{p^{2\ell +1+2\tau \theta _{2\ell +1}}-q^{2\ell +1+2\tau \theta _{2\ell +1}}}{p^{2\ell }(p^{s}-q^{s})} \biggr). \end{aligned}$$
$$\begin{aligned} \mathcal{P}_{s,p,q}^{\tau }(t;u) \geq &\frac{1}{[s-1]_{p,q}}+ \frac{q[s]_{p,q}}{[s-1]_{p,q}}\sum_{\ell =0}^{\infty } \mathcal{Q}_{s,p,q}(u) \biggl(\frac{p^{\ell +2\tau \theta _{\ell }}-q^{\ell +2\tau \theta _{\ell }}}{p^{\ell -1}(p^{s}-q^{s})} \biggr) \\ =&\frac{1}{[s-1]_{p,q}}+\frac{q[s]_{p,q}}{[s-1]_{p,q}}D_{s,p,q}(t;u) \\ =&\frac{1}{[s-1]_{p,q}}+\frac{q[s]_{p,q}}{[s-1]_{p,q}}u \end{aligned}$$
$$ \mathcal{P}_{s,p,q}^{\ell ,\tau }(t;u)\leq \frac{1}{[s-1]_{p,q}}+ \frac{[s]_{p,q}}{[s-1]_{p,q}}u. $$
Similarly, for \(f(t)=t^{2}\), we have
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
Similarly,
$$\begin{aligned} \mathcal{P}_{s,p,q}^{\tau }\bigl(t^{2};u\bigr) =& \sum_{\ell =0}^{\infty } \mathcal{Q}_{s,p,q}(u) \frac{1}{\mathcal{B}_{p,q}(\ell +2\tau \theta _{\ell }+1,s)}\int _{0}^{\infty } \frac{t^{\ell +2\tau \theta _{\ell }+2}}{(1\oplus pt)_{p,q}^{\ell +2\tau \theta _{\ell }+s+1}} \,\mathrm{d}_{p,q}t \\ =&\sum_{\ell =0}^{\infty }\mathcal{Q}_{s,p,q}(u) \frac{\mathcal{B}_{p,q}(\ell +2\tau \theta _{\ell }+3,s-2)}{\mathcal{B}_{p,q}(\ell +2\tau \theta _{\ell }+1,s)} \\ =&\sum_{\ell =0}^{\infty }\mathcal{Q}_{s,p,q}(u) \frac{\mathcal{B}_{p,q}(\ell +2\tau \theta _{\ell }+3,s-2)}{\mathcal{B}_{p,q}(\ell +2\tau \theta _{\ell }+1,s)} \\ =&\frac{1}{[s-1]_{p,q}[s-2]_{p,q}}\\ &{}\times\sum_{\ell =0}^{\infty } \mathcal{Q}_{s,p,q}(u)\frac{1}{p^{3+2\ell +4\tau \theta _{\ell }+1}}[\ell +2 \tau \theta _{\ell }+1]_{p,q}[\ell +2\tau \theta _{\ell }+2]_{p,q} \\ =&\frac{q^{3}[s]_{p,q}^{2}}{[s-1]_{p,q}[s-2]_{p,q}}\sum_{\ell =0}^{ \infty }\mathcal{Q}_{s,p,q}(u)\frac{1}{p^{5+4\tau \theta _{\ell }}} \biggl( \frac{p^{\ell +2\tau \theta _{\ell }}-q^{\ell +2\tau \theta _{\ell }}}{p^{\ell -1}(p^{s}-q^{s})} \biggr)^{2} \\ &{}+\frac{q(p+2q)[s]_{p,q}}{[s-1]_{p,q}[s-2]_{p,q}}\sum_{\ell =0}^{ \infty }\mathcal{Q}_{s,p,q}(u)\frac{1}{p^{4+2\tau \theta _{\ell }}} \biggl( \frac{p^{\ell +2\tau \theta _{\ell }}-q^{\ell +2\tau \theta _{\ell }}}{p^{\ell -1}(p^{s}-q^{s})} \biggr) \\ &{}+\frac{(p+q)}{p^{3}[s-1]_{p,q}[s-2]_{p,q}}\sum_{\ell =0}^{\infty } \mathcal{Q}_{s,p,q}(u). \end{aligned}$$
$$\begin{aligned} \mathcal{P}_{s,p,q}^{\tau }\bigl(t^{2};u\bigr) \geq &\frac{q^{3}[s]_{p,q}^{2}}{[s-1]_{p,q}[s-2]_{p,q}}\sum_{\ell =0}^{\infty } \mathcal{Q}_{s,p,q}(u) \biggl(\frac{p^{\ell +2\tau \theta _{\ell }}-q^{\ell +2\tau \theta _{\ell }}}{p^{\ell -1}(p^{s}-q^{s})} \biggr)^{2} \\ &{}+\frac{q(q+[2]_{p,q})[s]_{p,q}}{[s-1]_{p,q}[s-2]_{p,q}}\sum_{\ell =0}^{ \infty } \mathcal{Q}_{s,p,q}(u) \biggl( \frac{p^{\ell +2\tau \theta _{\ell }}-q^{\ell +2\tau \theta _{\ell }}}{p^{\ell -1}(p^{s}-q^{s})} \biggr) \\ &{}+\frac{[2]_{p,q}}{[s-1]_{p,q}[s-2]_{p,q}} \\ =&\frac{q^{3}[s]_{p,q}^{2}}{[s-1]_{p,q}[s-2]_{p,q}}D_{s,p,q}\bigl(t^{2};u\bigr)+ \frac{q(q+[2]_{p,q})[s]_{p,q}}{[s-1]_{p,q}[s-2]_{p,q}}D_{s,p,q}(t;u) \\ &{}+\frac{[2]_{p,q}}{[s-1]_{p,q}[s-2]_{p,q}}. \end{aligned}$$
$$\begin{aligned} \mathcal{P}_{s,p,q}^{\tau }\bigl(t^{2};u\bigr) \leq &\frac{[s]_{p,q}^{2}}{[s-1]_{p,q}[s-2]_{p,q}}D_{s,p,q}\bigl(t^{2};u\bigr)+ \frac{(1+[2]_{p,q})[s]_{p,q}}{[s-1]_{p,q}[s-2]_{p,q}}D_{s,p,q}(t;u) \\ &{}+\frac{[2]_{p,q}}{[s-1]_{p,q}[s-2]_{p,q}}. \end{aligned}$$
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
where \(\mathcal{M}_{f}\) is a constant depending on f, and σ is the weight function with \(\sigma (u)=1+u^{2}\). Moreover,
Note that \(C_{\sigma }[0,\infty )\) is a normed space with the norm given by \(\|f\|_{\sigma }=\sup_{u\geq 0 }\frac{|f(u)|}{\sigma (u)}\).
$$\begin{aligned}& \mathcal{L}:=\biggl\{ f:\lim_{u\rightarrow \infty }\frac{f(u)}{1+u^{2}}\text{ exits}\biggr\} , \\& B_{\sigma }[0,\infty ):= \bigl\{ f: \bigl\vert f(u) \bigr\vert \leq \mathcal{M}_{f}\sigma (u) \bigr\} , \end{aligned}$$
$$\begin{aligned}& C_{\sigma }[0,\infty ):=B_{\sigma }[0,\infty )\cap C[0,\infty ), \\& C_{\sigma }^{k}[0,\infty ):= \biggl\{ f:f\in C_{\sigma }[0,\infty ) \text{ and }\lim _{u\rightarrow \infty }\frac{f(u)}{\sigma (u)}=k< \infty \biggr\} . \end{aligned}$$
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
where \(f\in C[0,\infty )\cap \mathcal{L}\).
$$ \lim_{s\rightarrow \infty }\mathcal{P}_{s,p_{s},q_{s}}^{\tau }(f;u)=f(u), $$
Proof
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
$$ \lim_{s\rightarrow \infty } \bigl\Vert \mathcal{P}_{s,p_{s},q_{s}}^{\tau }(f)-f \bigr\Vert _{\sigma }=0. $$
(3.1)
Proof
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
$$ \lim_{s\rightarrow \infty } \bigl\Vert \mathcal{P}_{s,p_{s},q_{s}}^{\tau } ( 1 ) -1 \bigr\Vert _{\sigma }=0. $$
(3.2)
For \(\eta =1\),
Then
Similarly, if we take \(\eta =2\), then
This completes the proof. □
$$\begin{aligned} \bigl\Vert \mathcal{P}_{s,p_{s},q_{s}}^{\tau } ( t ) -u \bigr\Vert _{\sigma } & =\sup_{u\geq 0} \frac{ \vert \mathcal{P}_{s,p_{s},q_{s}}^{\tau }(t;u)-u \vert }{1+u^{2}} \\ & \leq \biggl( \frac{{}[ s]_{p_{s},q_{s}}}{[s-1]_{p_{s},q_{s}}}-1 \biggr) \sup _{u\geq 0}\frac{u}{1+u}+ \frac{1}{[s-1]_{p_{s},q_{s}}}\sup_{u\geq 0}\frac{1}{1+u}. \end{aligned}$$
$$ \lim_{s\rightarrow \infty } \bigl\Vert \mathcal{P}_{s,p_{s},q_{s}}^{\tau } ( t ) -u \bigr\Vert _{\sigma }=0. $$
(3.3)
$$\begin{aligned}& \begin{aligned} \bigl\Vert \mathcal{P}_{s,p_{s},q_{s}}^{\tau } \bigl( t^{2} \bigr) -u^{2} \bigr\Vert _{\sigma } ={}& \sup_{u\geq 0} \frac{ \vert \mathcal{P}_{s,p_{s},q_{s}}^{\tau }(t^{2};u)-u^{2} \vert }{1+u^{2}} \\ \leq {}& \biggl( \frac{[s]_{p_{s},q_{s}}^{2}}{[s-1]_{p_{s},q_{s}}[s-2]_{p,q}}-1 \biggr)\sup _{u\geq 0}\frac{u^{2}}{1+u^{2}} \\ &{}+\frac{[s]_{p_{s},q_{s}}}{[s-1]_{p_{s},q_{s}}[s-2]_{p_{s},q_{s}}} \bigl(1+[2]_{p_{s},q_{s}}+[1+2\tau ]_{p_{s},q_{s}} \bigr)\sup_{u\geq 0} \frac{u}{1+u^{2}} \\ &{}+\frac{[2]_{p_{s},q_{s}}}{[s-1]_{p_{s},q_{s}}[s-2]_{p,q}}\sup_{u \geq 0}\frac{1}{1+u^{2}}, \end{aligned} \\& \lim_{s\rightarrow \infty } \bigl\Vert \mathcal{P}_{s,p_{s},q_{s}}^{\tau } \bigl( t^{2} \bigr) -u^{2} \bigr\Vert _{\sigma }=0. \end{aligned}$$
(3.4)
Let
It is obvious that \(\lim_{\delta \rightarrow 0+}\omega _{\mu }(f;\delta )=0\) and for \(f\in C[0,\infty )\),
$$ \omega _{\mu }(f;\delta )=\sup_{ \vert t-u \vert \leq \delta }\sup _{u,t \in {}[ 0,\mu ]} \bigl\vert f(t)-f(u) \bigr\vert . $$
(3.5)
$$ \bigl\vert f(t)-f(u) \bigr\vert \leq \biggl( \frac{ \vert t-u \vert }{\delta }+1 \biggr) \omega _{\mu }(f;\delta ). $$
(3.6)
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
where \(\mathcal{C}_{f}\) is a constant depending only on f and \(\delta _{s}(u)=\sqrt{\mathcal{P}_{s,p_{s},q_{s}}^{\tau } (\Phi _{2};u )}\).
$$ \bigl\vert \mathcal{P}_{s,p_{s},q_{s}}^{\tau }(f;u)-f(u) \bigr\vert \leq 2\omega _{\mu +1} \bigl(f;\delta _{s}(u) \bigr)+6 \mathcal{C}_{f}\bigl(1+ \mu ^{2}\bigr) \bigl( \delta _{s}(u) \bigr) ^{2}, $$
Proof
For \(u\in {}[ 0,\mu ]\) and \(t\leq \mu +1\), with \(\mu >0\), we have
Furthermore, for any \(\delta >0\), \(u\in {}[ 0,\mu ]\), and \(t>\mu +1\), with \(\mu >0\),
From (3.7) and (3.8), we have
$$ \bigl\vert f(t)-f(u) \bigr\vert \leq \mathcal{C}_{f} \bigl( 2+u^{2}+t^{2} \bigr) \leq 6\mathcal{C}_{f} \bigl(1+\mu ^{2}\bigr) (t-u)^{2}. $$
(3.7)
$$ \bigl\vert f(t)-f(u) \bigr\vert \leq \omega _{\mu +1}\bigl(f; \vert t-u \vert \bigr) \leq \biggl( 1+ \frac{ \vert t-u \vert }{\delta } \biggr)\omega _{ \mu +1}(f;\delta ). $$
(3.8)
$$ \bigl\vert f(t)-f(u) \bigr\vert \leq 6 \mathcal{C}_{f}\bigl(1+\mu ^{2}\bigr) (t-u)^{2}+ \biggl( 1+\frac{ \vert t-u \vert }{\delta } \biggr)\omega _{\mu +1}(f; \delta ). $$
(3.9)
Applying operators \(\mathcal{P}_{s,p_{s},q_{s}}^{\tau }\) and the well-known Cauchy–Schwartz inequality, we have
Moreover, for any \(g\in C_{\sigma }[0,\infty )\), we know
Therefore,
where
$$\begin{aligned} \mathcal{P}_{s,p_{s},q_{s}}^{\tau } \bigl( \bigl\vert f(t)-f(u) \bigr\vert ;u \bigr) \leq &6\mathcal{C}_{f}\bigl(1+\mu ^{2}\bigr)\mathcal{P}_{s,p_{s},q_{s}}^{ \tau } ( \Phi _{2};u ) \\ &{}+\mathcal{P}_{s,p_{s},q_{s}}^{\tau } \biggl( 1+ \frac{ \vert t-u \vert }{\delta };u \biggr) \omega _{\mu +1}(f;\delta ). \end{aligned}$$
$$\begin{aligned} \mathcal{P}_{s,p_{s},q_{s}}^{\tau }(g;u)-g(u) =&\mathcal{P}_{s,p_{s},q_{s}}^{\tau }(g;u)-g(u) \mathcal{P}_{s,p_{s},q_{s}}^{\tau }(1;u) \\ =&\mathcal{P}_{s,p_{s},q_{s}}^{\tau } \bigl(g(t)-g(u);u \bigr) \\ \leq &\mathcal{P}_{s,p_{s},q_{s}}^{\tau } \bigl( \bigl\vert g(t)-g(u) \bigr\vert ;u \bigr). \end{aligned}$$
$$\begin{aligned} \bigl\vert \mathcal{P}_{s,p_{s},q_{s}}^{\tau }(f;u)-f(u) \bigr\vert \leq &6\mathcal{C}_{f}\bigl(1+\mu ^{2}\bigr)\mathcal{P}_{s,p_{s},q_{s}}^{\tau }(t-u)^{2} \\ &{}+ \biggl(1+\frac{1}{\delta }\mathcal{P}_{s,p_{s},q_{s}}^{\tau } \bigl( \vert t-u \vert ;u \bigr) \biggr)\omega _{\mu +1}(f;\delta ) \\ \leq &6\mathcal{C}_{f}\bigl(1+\mu ^{2}\bigr) \mathcal{P}_{s,p_{s},q_{s}}^{\tau } (\Phi _{2};u ) \\ &{}+ \biggl(1+\frac{1}{\delta }\mathcal{P}_{s,p_{s},q_{s}}^{\tau } ( \Phi _{2};u )^{\frac{1}{2}} \biggr)\omega _{\mu +1}(f; \delta ), \end{aligned}$$
$$ \mathcal{P}_{s,p_{s},q_{s}}^{\tau } \bigl( \vert t-u \vert ;u \bigr)\leq \mathcal{P}_{s,p_{s},q_{s}}^{\tau } (1;u )^{\frac{1}{2}}\mathcal{P}_{s,p_{s},q_{s}}^{\tau } \bigl((t-u)^{2};u \bigr)^{\frac{1}{2}}= \mathcal{P}_{s,p_{s},q_{s}}^{\tau } (\Phi _{2};u )^{\frac{1}{2}}. $$
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:
$$\begin{aligned}& K_{2}(f;\delta )=\inf_{u\geq 0} \bigl\{ \bigl( \Vert f-\psi \Vert _{C_{B}[0, \infty )}+\delta \bigl\Vert \psi ^{\prime \prime } \bigr\Vert _{C_{B}[0,\infty )} \bigr) :\psi \in C_{B}^{2}[0, \infty ) \bigr\} , \end{aligned}$$
(4.1)
$$\begin{aligned}& K_{2}(f;\delta )\leq \mathcal{C}\bigl\{ \omega _{2}(f; \sqrt{\delta })+\min (1, \delta ) \Vert f \Vert _{C_{B}[0,\infty )}\bigr\} , \\& \omega _{2}(f;\delta )=\sup_{0< v< \delta }\sup _{u\geq 0} \bigl\vert f(u+2v)-2f(u+v)+f(u) \bigr\vert . \end{aligned}$$
(4.2)
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
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.
$$ \bigl\vert \mathcal{R}_{s,p_{s},q_{s}}^{\tau }(\psi ;u)-\psi (u) \bigr\vert \leq \chi _{n}(u) \bigl\Vert \psi ^{\prime \prime } \bigr\Vert , $$
Proof
Let \(\psi \in C_{B}^{2}[0,\infty )\). We easily get \(\mathcal{R}_{s,p_{s},q_{s}}^{\tau }(1;u)=1\) and
Also
From the Taylor series expansion, we have
$$ \mathcal{R}_{s,p_{s},q_{s}}^{\tau }(t;u)=\mathcal{P}_{s,p_{s},q_{s}}^{ \tau }(t;u)+u- \biggl(\frac{[s]_{p_{s},q_{s}}u+1}{[s-1]_{p_{s},q_{s}}} \biggr)=u. $$
$$\begin{aligned}& \bigl\Vert \mathcal{P}_{s,p_{s},q_{s}}^{\tau }(f;u) \bigr\Vert \leq \Vert f \Vert , \\& \bigl\vert \mathcal{R}_{s,p_{s},q_{s}}^{\tau }(f;u) \bigr\vert \leq \bigl\vert \mathcal{P}_{s,p_{s},q_{s}}^{\tau }(f;u) \bigr\vert + \bigl\vert f(u) \bigr\vert - \biggl\vert f \biggl( \frac{[s]_{p_{s},q_{s}}u+1}{[s-1]_{p_{s},q_{s}}} \biggr) \biggr\vert \leq 3 \Vert f \Vert . \end{aligned}$$
(4.3)
$$ \psi (t)=\psi (u)+(t-u)\psi ^{\prime }(u)+ \int _{u}^{t}(t-\alpha ) \psi ^{\prime \prime }(\alpha )\,\mathrm{d}\alpha . $$
Applying the operator \(\mathcal{R}_{s,p_{s},q_{s}}^{\tau }\), we conclude that
Since
we conclude that
Hence,
Thus we complete the proof. □
$$\begin{aligned}& \begin{aligned} \mathcal{R}_{s,p_{s},q_{s}}^{\tau }(\psi ;u)-\psi (u)={}&\psi ^{ \prime }(u)\mathcal{R}_{s,p_{s},q_{s}}^{\tau }(t-u;u)+ \mathcal{R}_{s,p_{s},q_{s}}^{ \tau } \biggl( \int _{u}^{t}(t-\alpha )\psi ^{\prime \prime }( \alpha )\, \mathrm{d}\alpha ;u \biggr) \\ ={}&\mathcal{R}_{s,p_{s},q_{s}}^{\tau } \biggl( \int _{u}^{t}(t-\alpha ) \psi ^{\prime \prime }(\alpha )\,\mathrm{d}\alpha ;u \biggr) \\ ={}&\mathcal{P}_{s,p_{s},q_{s}}^{\tau } \biggl( \int _{u}^{t}(t-\alpha ) \psi ^{\prime \prime }(\alpha )\,\mathrm{d}\alpha ;u \biggr) \\ &{}- \int _{u}^{\frac{[s]_{p_{s},q_{s}}u+1}{[s-1]_{p_{s},q_{s}}}} \biggl( \frac{[s]_{p_{s},q_{s}}u+1}{[s-1]_{p_{s},q_{s}}}- \alpha \biggr)\psi ^{ \prime \prime }(\alpha )\,\mathrm{d}\alpha \end{aligned}\\& \begin{aligned} \bigl\vert \mathcal{R}_{s,p_{s},q_{s}}^{\tau }(\psi ;u)-\psi (u) \bigr\vert \leq{} & \biggl\vert \mathcal{P}_{s,p_{s},q_{s}}^{\tau } \biggl( \int _{u}^{t}(t- \alpha )\psi ^{\prime \prime }(\alpha )\,\mathrm{d}\alpha ;u \biggr) \biggr\vert \\ &{}+ \biggl\vert \int _{u}^{ \frac{[s]_{p_{s},q_{s}}u+1}{[s-1]_{p_{s},q_{s}}}} \biggl(\frac{[s]_{p_{s},q_{s}}u+1}{[s-1]_{p_{s},q_{s}}}-\alpha \biggr)\psi ^{ \prime \prime }(\alpha )\,\mathrm{d} \alpha \biggr\vert . \end{aligned} \end{aligned}$$
$$ \biggl\vert \int _{u}^{t}(t-\alpha )\psi ^{\prime \prime }( \alpha ) \,\mathrm{d}\alpha \biggr\vert \leq (t-u)^{2} \bigl\Vert \psi ^{\prime \prime } \bigr\Vert , $$
$$ \biggl\vert \int _{u}^{\frac{[s]_{p_{s},q_{s}}u+1}{[s-1]_{p_{s},q_{s}}}} \biggl(\frac{[s]_{p_{s},q_{s}}u+1}{[s-1]_{p_{s},q_{s}}}-\alpha \biggr)\psi ^{ \prime \prime }(\alpha )\,\mathrm{d} \alpha \biggr\vert \leq \biggl( \frac{[s]_{p_{s},q_{s}}u+1}{[s-1]_{p_{s},q_{s}}}-u \biggr)^{2} \bigl\Vert \psi ^{ \prime \prime } \bigr\Vert . $$
$$ \bigl\vert \mathcal{R}_{s,p_{s},q_{s}}^{\tau }(\psi ;u)-\psi (u) \bigr\vert \leq {\biggl\{ }\mathcal{P}_{s,p_{s},q_{s}}^{\tau } \bigl( (t-u)^{2};u \bigr) + \biggl(\frac{[s]_{p_{s},q_{s}}u+1}{[s-1]_{p_{s},q_{s}}}-u \biggr)^{2} {\biggr\} } \bigl\Vert \psi ^{\prime \prime } \bigr\Vert . $$
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
$$\begin{aligned} \bigl\vert \mathcal{P}_{s,p_{s},q_{s}}^{\tau }(f ;u)-f(u) \bigr\vert \leq &\mathcal{A} \biggl\{ \omega _{2} \biggl( f; \frac{\sqrt{\chi _{s}(u)}}{2} \biggr) +\min \biggl( 1, \frac{\chi _{s}(u)}{4} \biggr) \Vert f \Vert \biggr\} \\ &{}+\omega \bigl(f; \bigl\vert \mathcal{P}_{s,p_{s},q_{s}}(\Phi _{1};u) \bigr\vert \bigr). \end{aligned}$$
Proof
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,
By taking the infimum over all \(\psi \in C_{B}^{2}[0,\infty )\) and using (4.1), we get
□
$$\begin{aligned} \bigl\vert \mathcal{R}_{s,p_{s},q_{s}}^{\tau }(f ;u)-f(u) \bigr\vert =& \biggl\vert \mathcal{R}_{s,p_{s},q_{s}}^{\tau }(f ;u)-f(u)+f \biggl( \frac{[s]_{p_{s},q_{s}}u+1}{[s-1]_{p,q}} \biggr)-f(u) \biggr\vert \\ \leq & \bigl\vert \mathcal{R}_{s,p_{s},q_{s}}^{\tau }(f-\psi ;u) \bigr\vert + \bigl\vert \mathcal{R}_{s,p_{s},q_{s}}^{\tau }( \psi ;u)-\psi (u) \bigr\vert \\ &{}+ \bigl\vert \psi (u)-f(u) \bigr\vert + \biggl\vert f \biggl( \frac{[s]_{p_{s},q_{s}}u+1}{[s-1]_{p_{s},q_{s}}} \biggr)-f(u) \biggr\vert \\ \leq &4 \Vert f-\psi \Vert +\chi _{s}(u) \bigl\Vert \psi ^{ \prime \prime } \bigr\Vert \\ &{}+\omega \biggl(f; \biggl\vert \biggl( \frac{[s]_{p_{s},q_{s}}}{[s-1]_{p_{s},q_{s}}}-1 \biggr) u+ \frac{1}{[s-1]_{p_{s},q_{s}}} \biggr\vert \biggr). \end{aligned}$$
$$\begin{aligned} \bigl\vert \mathcal{R}_{s,p_{s},q_{s}}^{\tau }(f ;u)-f(u) \bigr\vert \leq &4K_{2} \biggl( f;\frac{\chi _{s}(u)}{4} \biggr) +\omega \biggl(f; \biggl\vert \biggl( \frac{[s]_{p_{s},q_{s}}}{[s-1]_{p_{s},q_{s}}}-1 \biggr) u+ \frac{1}{[s-1]_{p_{s},q_{s}}} \biggr\vert \biggr) \\ \leq &\mathcal{A} {\biggl\{ }\omega _{2} \biggl( f; \frac{\sqrt{\chi _{s}(u)}}{2} \biggr) +\min \biggl( 1;\frac{\chi _{s}(u)}{4} \biggr) \Vert f \Vert {\biggr\} } \\ &{}+\omega \biggl(f; \biggl\vert \biggl( \frac{[s]_{p_{s},q_{s}}}{[s-1]_{p_{s},q_{s}}}-1 \biggr) u+ \frac{1}{[s-1]_{p_{s},q_{s}}} \biggr\vert \biggr). \end{aligned}$$
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
$$ \mathrm{Lip}_{M}(\kappa )=\bigl\{ f: \bigl\vert f(t)-f(u) \bigr\vert \leq M \vert t-u \vert ^{ \kappa } \bigr\} . $$
(4.4)
Theorem 4.3
For all \(\kappa \in (0,1]\), \(s>2\), and \(f\in C_{B}[0,\infty )\), we have
where \(\delta _{s}(u)\) is given in Theorem 3.3.
$$ \bigl\vert \mathcal{P}_{s,p_{s},q_{s}}^{\tau }(f;u)-f(u) \bigr\vert \leq M \bigl( \delta _{s}(u) \bigr) ^{\kappa }, $$
Proof
We prove the claim by applying (4.4) and the well-known Hölder’s inequality:
This gives the desired result. □
$$\begin{aligned} \bigl\vert \mathcal{P}_{s,p_{s},q_{s}}^{\tau }(f;u)-f(u) \bigr\vert \leq &\mathcal{P}_{s,p_{s},q_{s}}^{\tau } \bigl( \bigl\vert f(t)-f(u) \bigr\vert ;u \bigr) \\ \leq &M| \mathcal{P}_{s,p_{s},q_{s}}^{\tau } \bigl( \vert t-u \vert ^{\kappa };u \bigr) \\ \leq &M \bigl( \mathcal{P}_{s,p_{s},q_{s}}^{\tau }(1;u) \bigr) ^{ \frac{2-\kappa }{2}} \bigl( \mathcal{P}_{s,p_{s},q_{s}}^{\tau }\bigl( \vert t-u \vert ^{2};u\bigr) \bigr) ^{\frac{\kappa }{2}} \\ =&M \bigl( \mathcal{P}_{s,p_{s},q_{s}}^{\tau }(\Phi _{2};u) \bigr) ^{ \frac{\kappa }{2}}. \end{aligned}$$
We denote
$$\begin{aligned}& C_{B}^{2}[0,\infty )= \bigl\{ \psi :\psi \in C_{B}[0,\infty ) \text{ and } \psi^{\prime },\psi ^{\prime \prime }\in C_{B}[0,\infty ) \bigr\} , \end{aligned}$$
(4.5)
$$\begin{aligned}& \Vert \psi \Vert _{C_{B}^{2}[0,\infty )}= \Vert \psi \Vert _{C_{B}[0,\infty )}+ \bigl\Vert \psi ^{\prime } \bigr\Vert _{C_{B}[0, \infty )}+ \bigl\Vert \psi ^{\prime \prime } \bigr\Vert _{C_{B}[0,\infty )}, \end{aligned}$$
(4.6)
$$\begin{aligned}& \Vert \psi \Vert _{C_{B}[0,\infty )}=\sup_{u\geq 0} \bigl\vert \psi (u) \bigr\vert . \end{aligned}$$
(4.7)
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
where \(\Theta _{s}(u)=\sqrt{\delta _{s}(u)}+ \frac{ (\delta _{s}(u) )^{2}}{2}\).
$$ \bigl\vert \mathcal{P}_{s,p_{s},q_{s}}^{\tau }(\psi ;u)-\psi (u) \bigr\vert \leq \Theta _{s}(u) \Vert \psi \Vert _{C_{B}^{2}[0,\infty )}, $$
(4.8)
Proof
Let \(\psi \in C_{B}^{2}[0,\infty )\). Then
where if we take
then we have
Therefore
$$ \psi (t)=\psi (u)+\psi ^{\prime }(u) (t-u)+\psi ^{\prime \prime }( \varphi ) \frac{(t-u)^{2}}{2}\quad \text{for } \varphi \in (u,t), $$
$$\begin{aligned}& \mathcal{S}=\sup_{u\geq 0} \bigl\vert \psi ^{\prime }(u) \bigr\vert = \bigl\Vert \psi ^{\prime } \bigr\Vert _{C_{B}[0,\infty )}\leq \Vert \psi \Vert _{C_{B}^{2}[0, \infty )}, \\& \mathcal{T}=\sup_{u\geq 0} \bigl\vert \psi ^{\prime \prime }(u) \bigr\vert = \bigl\Vert \psi ^{\prime \prime } \bigr\Vert _{C_{B}[0,\infty )} \leq \Vert \psi \Vert _{C_{B}^{2}[0, \infty )}, \end{aligned}$$
$$\begin{aligned} \bigl\vert \psi (t)-\psi (u) \bigr\vert \leq & \mathcal{S} \vert t-u \vert +\frac{1}{2}\mathcal{T}(t-u)^{2} \\ \leq & \biggl( \vert t-u \vert +\frac{1}{2}(t-u)^{2} \biggr) \Vert \psi \Vert _{C_{B}^{2}[0,\infty )}. \end{aligned}$$
$$\begin{aligned} \bigl\vert \mathcal{P}_{s,p_{s},q_{s}}^{\tau }(\psi ;u)-\psi (u) \bigr\vert \leq & \biggl( \mathcal{P}_{s,p_{s},q_{s}}^{\tau }\bigl( \vert t-u \vert ;u\bigr)+\frac{1}{2}\mathcal{P}_{s,p_{s},q_{s}}^{\tau } \bigl((t-u)^{2};u \bigr) \biggr) \Vert \psi \Vert _{C_{B}^{2}[0,\infty )} \\ \leq & \biggl( \bigl(\mathcal{P}_{s,p_{s},q_{s}}^{\tau } ( \Phi _{2};u ) \bigr)^{\frac{1}{2}}+\frac{1}{2} \mathcal{P}_{s,p_{s},q_{s}}^{\tau } (\Phi _{2};u ) \biggr) \Vert \psi \Vert _{C_{B}^{2}[0,\infty )}. \end{aligned}$$
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.
Acknowledgements
NA
Competing interests
The authors declare that they have no competing interests.
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