Modified (p,q)-Bernstein-Schurer operators and their approximation properties

Abstract

In this paper, we introduce modified (p, q)-Bernstein–Schurer operators and discuss their statistical approximation properties based on Korovkins type approximation theorem. We compute the rate of convergence and also prove a Voronovskaja-type theorem.

Keywords:

• q-integers
• (p
• q)-integers
• Bernstein operator
• (p
• q)-Bernstein operator
• (p
• q)-Bernstein–Schurer operator
• modulus of continuity
• Korovkins approximation theorem

AMS Subject Classifications:

• 41A10
• 41A25
• 41A36

Public Interest Statement

In this paper, we have modified the (p, q)-Bernstein–Schurer operators and discussed their statistical approximation properties based on Korovkins type approximation theorem. We have also established the rate of convergence of these operators using the modulus of continuity. Furthermore, we have proved a Voronovskaja-type theorem. One of its advantages of using the extra parameter p has been mentioned in Mursaleen, Faisal Khan and Asif Khan (2016) to study (p, q)-approximation by Lorentz operators in compact disk. Another nice application has been given by Khan, Lobiyal and Kilicman (2015) and Khan and Lobiyal (2015) in computer-aided geometric design and applied these Bernstein bases for construction of (p, q)-Bézier curves and surfaces based on (p, q)-integers.

1. Introduction and preliminaries

In Lupaş (1987) introduced the first q-analogue of the classical Bernstein operators and investigated its approximating and shape-preserving properties. Another q-generalization of the classical Bernstein polynomial is due to Phillips (1997). Several generalizations of well-known positive linear operators based on q-integers were introduced and their approximation properties have been studied by several researchers.

Recently, Mursaleen et al. introduced (p, q)-calculus in approximation theory and constructed the (p, q)-analogue of Bernstein operators Mursaleen, Ansari, and Khan (2015a) and (p, q)-analogue of Bernstein–Stancu operators (Mursaleen, Ansari, & Khan, 2015b). Most recently, the (p, q)-analogue of some more operators has been studied in Acar (2010), Acar, Aral, and Mohiuddine (2016a,2016b), Cai and Zhou (2016), Mursaleen, Alotaibi, and Ansari (2016), Mursaleen and Nasiruzzaman (2016), Mursaleen, Nasiuzzaman, and Nurgali (2015) and Mursaleen and Nasiruzzaman (2015). One of its advantages of using the extra parameter p has been mentioned in Mursaleen, Khan, and Khan (2016) to study (p, q)-approximation by Lorentz operators in compact disk. Another nice application has been given by Khan et al. (2015) and Khan and Lobiyal (2015) in computer-aided geometric design and applied these Bernstein bases for construction of (p, q)-Bézier curves and surfaces based on (p, q)-integers.

The (p, q)-integer was introduced to generalize or unify several forms of q-oscillator algebras well known in the Physics literature related to the representation theory of single-parameter quantum algebras. The (p, q)-integer is defined by(1.1) $$\begin{array}{c}\hfill {\left[n\right]}_{p,q}=p^{n-2}+q{p}^{n-3}+\cdots +q^{n-2}=\left\{\begin{array}{cc}{\displaystyle \frac{p^n-q^n}{p-q}}\hfill & (p\ne q\ne 1)\hfill \\ {\displaystyle \frac{1-q^n}{1-q}}\hfill & (p=1)\hfill \\ n\hfill & (p=q=1)\hfill \end{array}\right.\end{array}$$(1.1)

where $0<q<p\le 1.$

The (p, q)-binomial expansion is$$\begin{array}{cc}\hfill {(ax+by)}_{p,q}^n& :=\sum _{k=0}^n{p}^{\frac{(n-k)(n-k-1)}{2}}{q}^{\frac{k(k-1)}{2}}{\left[\begin{array}{c}n\\ k\end{array}\right]}_{p,q}{a}^{n-k}{b}^k{x}^{n-k}{y}^k,\hfill \\ \hfill {(x+y)}_{p,q}^n& :=(x+y)(px+qy)(p^{2}x+q^{2}y)\cdots (p^{n-1}x+q^{n-1}y),\hfill \\ \hfill {(1-x)}_{p,q}^n& :=(1-x)(p-qx)(p^2-q^{2}x)\cdots (p^{n-1}-q^{n-1}x).\hfill \end{array}$$

The (p, q)-binomial coefficients are defined by$$\begin{array}{c}\hfill {\left[\begin{array}{c}n\\ k\end{array}\right]}_{p,q}:=\frac{{[n]}_{p,q}!}{{[k]}_{p,q}!{[n-k]}_{p,q}!}.\end{array}$$

In Schurer (1962) introduced and studied the operators $C_{m,d}:C[0,d+1]\to CC[0,1])$ defined for any $m\in \mathbb{N}$ and d be fixed in $\mathbb{N}$ and any function $f\in C[0,d+1]$ as follows:(1.2) $$\begin{array}{c}\hfill C_{m,d}(f;x)=\sum _{k=0}^{m+d}\left[\begin{array}{c}m+d\\ k\end{array}\right]x^k{(1-x)}^{m+d-k}f\left(\frac{k}{m}\right),x\in [0,1].\end{array}$$(1.2)

In Muraru (2011) constructed the q-Bernstein–Schurer operators defined by(1.3) $$\begin{array}{c}\hfill {\tilde{B}}_{n,p}(f;q;x)=\sum _{k=0}^{n+p}{\left[\begin{array}{c}n+p\\ k\end{array}\right]}_q{x}^k\prod _{s=0}^{n+p-k-1}(1-q^{s}x)f\left(\frac{{[k]}_q}{{[n]}_q}\right),x\in [0,1].\end{array}$$(1.3)

Mursaleen et al. (2015) introduced the generalized (p, q)-analogue of Bernstein–Schurer operators as follows:(1.4) $$\begin{array}{c}\hfill {S^{\ast }}_{n,p,q}(f;x)=\frac{1}{p^{\frac{(n+\ell )(n+\ell -1)}{2}}}\sum _{k=0}^{n+\ell }{\left[\begin{array}{c}n+\ell \\ k\end{array}\right]}_{p,q}{p}^{\frac{k(k-1)}{2}}{x}^k\prod _{s=0}^{n+\ell -k-1}(p^s-q^{s}x)f\left(\frac{{[k]}_{p,q}}{p^{k-n-\ell }{[n]}_{p,q}}\right)\end{array}$$(1.4) $x\in [0,1]$.

2. Construction of operators

We consider $0<q<p\le 1$ and for any $m\in \mathbb{N},f\in C[0,d+1]$, d is fixed and $d\in \mathbb{N}\cup \{0\}$. We define the modified (p, q)-Bernstein–Schurer operators for $x\in [0,1]$ as follows: $L_{m,d}^{p,q}(f;x)=$(2.1) $$\begin{array}{c}\hfill \frac{1}{p^{\frac{(m+d)(m+d-1)}{2}}}\sum _{k=0}^{m+d}{\left[\begin{array}{c}m+d\\ k\end{array}\right]}_{p,q}{(r_{m,d}(p,q;x))}_{p,q}^k\prod _{s=0}^{m+d-k-1}(p^s-q^s{r}_{m,d}(p,q;x))f\left(\frac{{[k]}_{p,q}}{p^{k-m-d}{[m]}_{p,q}}\right).\end{array}$$(2.1)

where $r_{m,d}(p,q;x)=\frac{{[m]}_{p,q}x}{{[m+d]}_{p,q}}$. In case of $p=1$, the operators turn out the modified q-Schurer operators defined in Mursaleen et al. (mnn) and if we replace $r_{m,d}(q;x)$ by x, then we get (1.3). Moreover, if we take $d=0$ and $r_{m,d}(p,q;x)=x$, we get (p, q)-Bernstein operators defined in Mursaleen et al. (2015a).

Lemma 2.1

Let $L_{m,d}^{p,q}(f;x)$ be the operators defined by (2.1). Then, for any function $f\in C[0,d+1]$, $d\in \mathbb{N}\cup \{0\},x\in [0,1],$ we have

(i)	$L_{m,d}^{p,q}(1;x)=1$,
(ii)	$L_{m,d}^{p,q}(t;x)=x$,
(iii)	$L_{m,d}^{p,q}(t^2;x)=\frac{{[m+d]}_{p,q}}{{[m]}_{p,q}^2}\left({[m+d]}_{p,q}{r}_{m,d}^2(p,q;x)+p^{m+d-1}{r}_{m,d}(p,q;x)\left(1-r_{m,d}(p,q;x)\right)\right)$,
(iv)	$$\begin{array}{cc}\hfill L_{m,d}^{p,q}(t^3;x)& =\frac{p^{2(m+d-1)}{[m+d]}_{p,q}}{{[m]}_{p,q}^3}{r}_{m,d}(p,q;x)\hfill \\ \hfill & +\frac{2p^{m+d-1}q+p^{m+d-2}{q}^2}{{[m]}_{p,q}^3}{[m+d]}_{p,q}{[m+d-1]}_{p,q}{r}_{m,d}^2(p,q;x)\hfill \\ \hfill & +\frac{q^3}{{[m]}_{p,q}^3}{[m+d]}_{p,q}{[m+d-1]}_{p,q}{[m+d-2]}_{p,q}{r}_{m,d}^3(p,q;x)\text{for}m+d\ge 2,\hfill \end{array}$$
(v)	$$\begin{array}{cc}\hfill L_{m,d}^{p,q}(t^4;x)& =\frac{p^{3(m+d-1)}{[m+d]}_{p,q}}{{[m]}_{p,q}^4}{r}_{m,d}(p,q;x)\hfill \\ \hfill & +\frac{3p^{2(m+d-1)}q+3p^{2m+2d-3}{q}^2+p^{m+d-2}{q}^3}{{[m]}_{p,q}^4}{[m+d]}_{p,q}{[m+d-1]}_{p,q}{r}_{m,d}^2(p,q;x)\hfill \\ \hfill & +\frac{3p^{m+d-1}{q}^3+2p^{m+d-2}{q}^4+p^{m+d-3}{q}^5}{{[m]}_{p,q}^4}{[m+d]}_{p,q}{[m+d-1]}_{p,q}{[m+d-2]}_{p,q}{r}_{m,d}^3(p,q;x)\hfill \\ \hfill & +\frac{q^6}{{[m]}_{p,q}^4}{[m+d]}_{p,q}{[m+d-1]}_{p,q}{[m+d-2]}_{p,q}{[m+d-3]}_{p,q}{r}_{m,d}^4(p,q;x)\text{for}m+d\ge 3.\hfill \end{array}$$

Proof

 

(i)	For $0<q<p\le 1$, we use the known identity from Mursaleen et al. (2015a) $$\begin{array}{c}\hfill \sum _{k=0}^m{\left[\begin{array}{c}m\\ k\end{array}\right]}_{p,q}{(r_{m,d}(p,q;x))}_{p,q}^k\prod _{s=0}^{m-k-1}(p^s-q^s{r}_{m,d}(p,q;x))=p^{\frac{(m)(m-1)}{2}}.\end{array}$$ We have $$\begin{array}{c}\hfill (1-{(r_{m,d}(p,q;x))}_{p,q}^{m+d-k}=\prod _{s=0}^{m+d-k-1}(p^s-q^s{r}_{m,d}(p,q;x)),\end{array}$$ and $$\begin{array}{c}\hfill \sum _{k=0}^{m+d}{\left[\begin{array}{c}m+d\\ k\end{array}\right]}_{p,q}{(r_{m,d}(p,q;x))}_{p,q}^k\prod _{s=0}^{m+d-k-1}(p^s-q^s{r}_{m,d}(p,q;x))=p^{\frac{(m+d)(m+d-1)}{2}},\end{array}$$ Consequently, we have $L_{m,d}^{p,q}(1;x)=1$.
(ii)	Using ${(r_{m,d}(p,q;x))}_{p,q}^{k+1}=p^k\frac{{[m]}_{p,q}}{{[m+d]}_{p,q}}x{(r_{m,d}(p,q;x))}_{p,q}^k,$ we have $L_{m,d}^{p,q}(t;x)$$$\begin{array}{cc}& =\frac{1}{p^{\frac{(m+d)(m+d-1)}{2}}}\sum _{k=0}^{m+d}{\left[\begin{array}{c}m+d\\ k\end{array}\right]}_{p,q}{(r_{m,d}(p,q;x))}_{p,q}^k\prod _{s=0}^{m+d-k-1}(p^s-q^s{r}_{m,d}(p,q;x))\frac{{[k]}_{p,q}}{p^{k-m-d}{[m]}_{p,q}}\hfill \\ & =\frac{{[m+d]}_{p,q}}{p^{\frac{(m+d)(m+d-1)}{2}-m-d}}\sum _{k=0}^{m+d-1}{\left[\begin{array}{c}m+d-1\\ k\end{array}\right]}_{p,q}{\left(\frac{{[m]}_{p,q}x}{{[m+d]}_{p,q}}\right)}_{p,q}^{k+1}\prod _{s=0}^{m+d-k-2}\left(p^s-q^s\frac{{[m]}_{p,q}x}{{[m+d]}_{p,q}}\right)\frac{1}{p^{k+1}{[m]}_{p,q}}\hfill \\ & =\frac{p^k{[m]}_{p,q}{[m+d]}_{p,q}x}{p^k{p}^{\frac{(m+d)(m+d-1)}{2}-m-d+1}{[m]}_{p,q}{[m+d]}_{p,q}}\sum _{k=0}^{m+d-1}{\left[\begin{array}{c}m+d-1\\ k\end{array}\right]}_{p,q}{\left(\frac{{[m]}_{p,q}x}{{[m+d]}_{p,q}}\right)}_{p,q}^k\hfill \\ & \prod _{s=0}^{m+d-k-2}\left(p^s-q^s\frac{{[m]}_{p,q}x}{{[m+d]}_{p,q}}\right)\hfill \\ & =x.\hfill \end{array}$$
(iii)	Using ${(r_{m,d}(p,q;x))}_{p,q}^{k+2}=p^{k+1}\frac{{[m]}_{p,q}}{{[m+d]}_{p,q}}x{(r_{m,d}(p,q;x))}_{p,q}^{k+1}$, ${[k+1]}_{p,q}=p^k+q{[k]}_{p,q}$ and $q{[m+d-1]}_{p,q}={[m+d]}_{p,q}-p^{m+d-1}$, we have $$\begin{array}{cc}\hfill L_{m,d}^{p,q}(t^2;x)& =\frac{1}{p^{\frac{(m+d)(m+d-1)}{2}}}\sum _{k=0}^{m+d}{\left[\begin{array}{c}m+d\\ k\end{array}\right]}_{p,q}{\left(\frac{{[m]}_{p,q}x}{{[m+d]}_{p,q}}\right)}_{p,q}^k\prod _{s=0}^{m+d-k-1}\left(p^s-q^s\frac{{[m]}_{p,q}x}{{[m+d]}_{p,q}}\right)\hfill \\ & \frac{{[k]}_{p,q}^2}{p^{2k-2m-2d}{[m]}_{p,q}^2}\hfill \\ & =\frac{1}{p^{\frac{(m+d)(m+d-1)}{2}-2m-2d}}\frac{{[m+d]}_{p,q}}{{[m]}_{p,q}^2}\sum _{k=1}^{m+d-1}{\left[\begin{array}{c}m+d-1\\ k-1\end{array}\right]}_{p,q}\frac{1}{p^{2k}}{\left(\frac{{[m]}_{p,q}x}{{[m+d]}_{p,q}}\right)}_{p,q}^k\hfill \\ & \prod _{s=0}^{m+d-k-1}\left(p^s-q^s\frac{{[m]}_{p,q}x}{{[m+d]}_{p,q}}\right){[k]}_{p,q}\hfill \\ & =\frac{1}{p^{\frac{(m+d)(m+d-1)}{2}-2m-2d+2}}\frac{{[m+d]}_{p,q}}{{[m]}_{p,q}^2}\sum _{k=0}^{m+d-1}{\left[\begin{array}{c}m+d-1\\ k\end{array}\right]}_{p,q}\frac{1}{p^k}\left(\frac{{[m]}_{p,q}x}{{[m+d]}_{p,q}}\right){\left(\frac{{[m]}_{p,q}x}{{[m+d]}_{p,q}}\right)}_{p,q}^k\hfill \\ & \prod _{s=0}^{m+d-k-2}\left(p^s-q^s\frac{{[m]}_{p,q}x}{{[m+d]}_{p,q}}\right)\left(p^k+q{[k]}_{p,q}\right)\hfill \\ & =\frac{1}{p^{\frac{(m+d)(m+d-1)}{2}-2m-2d+2}}\frac{x}{{[m]}_{p,q}}\sum _{k=0}^{m+d-1}{\left[\begin{array}{c}m+d-1\\ k\end{array}\right]}_{p,q}{\left(\frac{{[m]}_{p,q}x}{{[m+d]}_{p,q}}\right)}_{p,q}^k\hfill \\ & \prod _{s=0}^{m+d-k-2}\left(p^s-q^s\frac{{[m]}_{p,q}x}{{[m+d]}_{p,q}}\right)\hfill \\ & +\frac{1}{p^{\frac{(m+d)(m+d-1)}{2}-2m-2d+2}}\frac{q{[m+d-1]}_{p,q}x}{{[m]}_{p,q}}\sum _{k=0}^{m+d-2}{\left[\begin{array}{c}m+d-2\\ k\end{array}\right]}_{p,q}\frac{1}{p^{k+1}}{\left(\frac{{[m]}_{p,q}x}{{[m+d]}_{p,q}}\right)}_{p,q}^{k+1}\hfill \\ & \prod _{s=0}^{m+d-k-3}\left(p^s-q^s\frac{{[m]}_{p,q}x}{{[m+d]}_{p,q}}\right)\hfill \\ & =\frac{1}{p^{\frac{(m+d)(m+d-1)}{2}-2m-2d+2}}{p}^{\frac{(m+d-1)(m+d-2)}{2}}\frac{x}{{[m]}_{p,q}}+\frac{1}{p^{\frac{(m+d-2)(m+d-3)}{2}}}\frac{q{[m+d-1]}_{p,q}{x}^2}{{[m+d]}_{p,q}}\sum _{k=0}^{m+d-2}\hfill \\ & {\left[\begin{array}{c}m+d-2\\ k\end{array}\right]}_{p,q}{\left(\frac{{[m]}_{p,q}x}{{[m+d]}_{p,q}}\right)}_{p,q}^k\prod _{s=0}^{m+d-k-3}\left(p^s-q^s\frac{{[m]}_{p,q}x}{{[m+d]}_{p,q}}\right)\hfill \\ & =\frac{p^{m+d-1}x}{{[m]}_{p,q}}+\frac{q{[m+d-1]}_{p,q}}{{[m+d]}_{p,q}}{x}^2\hfill \\ & =\frac{p^{m+d-1}x}{{[m]}_{p,q}}+\left(\frac{{[m+d]}_{p,q}-p^{m+d-1}}{{[m+d]}_{p,q}}\right)x^2\hfill \\ & =\frac{p^{m+d-1}{[m+d]}_{p,q}}{{[m]}_{p,q}^2}{r}_{m,d}(p,q;x)+x^2-\frac{p^{m+d-1}{[m+d]}_{p,q}}{{[m]}_{p,q}^2}{r}_{m,d}^2(p,q;x)\hfill \\ & =x^2+\frac{p^{m+d-1}{[m+d]}_{p,q}}{{[m]}_{p,q}^2}{r}_{m,d}(p,q;x)\left(1-r_{m,d}(p,q;x))\right)\hfill \\ & =\frac{{[m+d]}_{p,q}}{{[m]}_{p,q}^2}\left({[m+d]}_{p,q}{r}_{m,d}^2(p,q;x)+p^{m+d-1}{r}_{m,d}(p,q;x)\left(1-r_{m,d}(p,q;x)\right)\right).\hfill \end{array}$$
(iv)	we have $$\begin{array}{cc}\hfill L_{m,d}^{p,q}(t^3;x)& =\frac{1}{p^{\frac{(m+d)(m+d-1)}{2}}}\sum _{k=0}^{m+d}{\left[\begin{array}{c}m+d\\ k\end{array}\right]}_{p,q}{\left(\frac{{[m]}_{p,q}x}{{[m+d]}_{p,q}}\right)}_{p,q}^k\prod _{s=0}^{m+d-k-1}\left(p^s-q^s\frac{{[m]}_{p,q}x}{{[m+d]}_{p,q}}\right)\hfill \\ & \frac{{[k]}_{p,q}^3}{p^{3k-3m-3d}{[m]}_{p,q}^3}\hfill \\ & =\frac{1}{p^{\frac{(m+d)(m+d-1)}{2}-3m-3d+3}}x\sum _{k=0}^{m+d-1}{\left[\begin{array}{c}m+d-1\\ k\end{array}\right]}_{p,q}{(\frac{{[m]}_{p,q}x}{{[m+d]}_{p,q}})}_{p,q}^k\prod _{s=0}^{m+d-k-2}\hfill \\ & \left(p^s-q^s\frac{{[m]}_{p,q}x}{{[m+d]}_{p,q}}\right)\frac{(p^{2k}+2p^{k}q{[k]}_{p,q}+q^2{[k]}_{p,q}^2)}{p^{2k}{[m]}_{p,q}^2}\hfill \\ & =\frac{p^{2(m+d-1)}}{{[m]}_{p,q}^2}x+\frac{2p^{m+d-1}q+p^{m+d-2}{q}^2}{{[m]}_{p,q}{[m+d]}_{p,q}}{[m+d-1]}_{p,q}{x}^2\hfill \\ & +\frac{q^3}{{[m+d]}_{p,q}^2}{[m+d-1]}_{p,q}{[m+d-2]}_{p,q}{x}^3\hfill \\ & =\frac{p^{2(m+d-1)}{[m+d]}_{p,q}}{{[m]}_{p,q}^3}{r}_{m,d}(p,q;x)\hfill \\ & +\frac{2p^{m+d-1}q+p^{m+d-2}{q}^2}{{[m]}_{p,q}^3}{[m+d]}_{p,q}{[m+d-1]}_{p,q}{r}_{m,d}^2(p,q;x)\hfill \\ & +\frac{q^3}{{[m]}_{p,q}^3}{[m+d]}_{p,q}{[m+d-1]}_{p,q}{[m+d-2]}_{p,q}{r}_{m,d}^3(p,q;x).\hfill \end{array}$$
(v)	we have $$\begin{array}{cc}\hfill L_{m,d}^{p,q}(t^4;x)& =\frac{1}{p^{\frac{(m+d)(m+d-1)}{2}}}\sum _{k=0}^{m+d}{\left[\begin{array}{c}m+d\\ k\end{array}\right]}_{p,q}{\left(\frac{{[m]}_{p,q}x}{{[m+d]}_{p,q}}\right)}_{p,q}^k\prod _{s=0}^{m+d-k-1}\left(p^s-q^s\frac{{[m]}_{p,q}x}{{[m+d]}_{p,q}}\right)\hfill \\ & \frac{{[k]}_{p,q}^4}{p^{4k-4m-4d}{[m]}_{p,q}^4}\hfill \\ & =\frac{1}{p^{\frac{(m+d)(m+d-1)}{2}-4m-4d+4}}x\sum _{k=0}^{m+d-1}{\left[\begin{array}{c}m+d-1\\ k\end{array}\right]}_{p,q}{\left(\frac{{[m]}_{p,q}x}{{[m+d]}_{p,q}}\right)}_{p,q}^k\prod _{s=0}^{m+d-k-2}\hfill \\ & \left(p^s-q^s\frac{{[m]}_{p,q}x}{{[m+d]}_{p,q}}\right)\frac{(p^{3k}+3p^{2k}q{[k]}_{p,q}+3p^k{q}^2{[k]}_{p,q}^2+q^3{[k]}_{p,q}^3)}{p^{3k}{[m]}_{p,q}^3}\hfill \\ & =\frac{p^{3(m+d-1)}}{{[m]}_{p,q}^3}x+\frac{3p^{2(m+d-1)}q}{{[m]}_{p,q}^2{[m+d]}_{p,q}}{[m+d-1]}_{p,q}{x}^2+\frac{3q^2{[m+d-1]}_{p,q}}{p^{\frac{(m+d)(m+d-1)}{2}-4m-4d+4}}{x}^2\sum _{k=0}^{m+d-2}\hfill \\ & {\left[\begin{array}{c}m+d-2\\ k\end{array}\right]}_{p,q}{\left(\frac{{[m]}_{p,q}x}{{[m+d]}_{p,q}}\right)}_{p,q}^k\prod _{s=0}^{m+d-k-3}\left(p^s-q^s\frac{{[m]}_{p,q}x}{{[m+d]}_{p,q}}\right)\frac{(p^k+q{[k]}_{p,q})}{p^{k+2}{[m]}_{p,q}^2{[m+d]}_{p,q}}\hfill & \hfill +\frac{q^3{[m+d-1]}_{p,q}}{p^{\frac{(m+d)(m+d-1)}{2}-4m-4d+4}}{x}^2\sum _{k=0}^{m+d-2}{\left[\begin{array}{c}m+d-2\\ k\end{array}\right]}_{p,q}{\left(\frac{{[m]}_{p,q}x}{{[m+d]}_{p,q}}\right)}_{p,q}^k\prod _{s=0}^{m+d-k-3}\\ & \left(p^s-q^s\frac{{[m]}_{p,q}x}{{[m+d]}_{p,q}}\right)\frac{(p^{2k}+2p^{k}q{[k]}_{p,q}+q^2{[k]}_{p,q}^2)}{p^{2k+3}{[m]}_{p,q}^2{[m+d]}_{p,q}}\hfill \\ & =\frac{p^{3(m+d-1)}}{{[m]}_{p,q}^3}x+\frac{3p^{2(m+d-1)}q+3p^{2m+2d-3}{q}^2+p^{m+d-2}{q}^3}{{[m]}_{p,q}^2{[m+d]}_{p,q}}{[m+d-1]}_{p,q}{x}^2\hfill \\ & +\frac{3p^{m+d-1}{q}^3+2p^{m+d-2}{q}^4+p^{m+d-3}{q}^5}{{[m]}_{p,q}{[m+d]}_{p,q}^2}{[m+d-1]}_{p,q}{[m+d-2]}_{p,q}{x}^3\hfill \\ & +\frac{q^6}{{[m+d]}_{p,q}^3}{[m+d-1]}_{p,q}{[m+d-2]}_{p,q}{[m+d-3]}_{p,q}{x}^4\hfill & \hfill =\frac{p^{3(m+d-1)}{[m+d]}_{p,q}}{{[m]}_{p,q}^4}{r}_{m,d}(p,q;x)\\ & +\frac{3p^{2(m+d-1)}q+3p^{2m+2d-3}{q}^2+p^{m+d-2}{q}^3}{{[m]}_{p,q}^4}{[m+d]}_{p,q}{[m+d-1]}_{p,q}{r}_{m,d}^2(p,q;x)\hfill \\ & +\frac{3p^{m+d-1}{q}^3+2p^{m+d-2}{q}^4+p^{m+d-3}{q}^5}{{[m]}_{p,q}^4}{[m+d]}_{p,q}{[m+d-1]}_{p,q}{[m+d-2]}_{p,q}{r}_{m,d}^3(p,q;x)\hfill \\ & +\frac{q^6}{{[m]}_{p,q}^4}{[m+d]}_{p,q}{[m+d-1]}_{p,q}{[m+d-2]}_{p,q}{[m+d-3]}_{p,q}{r}_{m,d}^4(p,q;x).\hfill \end{array}$$

$\square$

Lemma 2.2

Let $0<q<p\le 1$, and for any $m\in \mathbb{N}$, we have

(i)	$L_{m,d}^{p,q}(t-x;x)=0,$
(ii)	$L_{m,d}^{p,q}({(t-x)}^2;x)=\frac{p^{m+d-1}}{{[m]}_{p,q}}x\left(1-\frac{{[m]}_{p,q}x}{{[m+d]}_{p,q}}\right)\le \frac{p^{m+d-1}}{{[m]}_{p,q}}\left(1-\frac{{[m]}_{p,q}x}{{[m+d]}_{p,q}}\right),$
(iii)	$$\begin{array}{cc}\hfill L_{m,d}^{p,q}({(t-x)}^3;x)& =L_{m,d}^{p,q}(t^3;x)-3x{L}_{m,d}^{p,q}(t^2;x)+3x^2{L}_{m,d}^{p,q}(t;x)-x^3\hfill \\ \hfill & =\frac{p^{2(m+d-1)}}{{[m]}_{p,q}^2}x-\left(\frac{3p^{m+d-1}}{{[m]}_{p,q}}-\frac{2p^{m+d-1}q+p^{m+d-2}{q}^2}{{[m]}_{p,q}{[m+d]}_{p,q}}{[m+d-1]}_{p,q}\right)x^2\hfill \\ \hfill & +\left(\frac{q^3}{{[m+d]}_{p,q}^2}{[m+d-1]}_{p,q}{[m+d-2]}_{p,q}-\frac{3q{[m+d-1]}_{p,q}}{{[m+d]}_{p,q}}+2\right)x^3,\hfill \end{array}$$
(iv)	$$\begin{array}{cc}& L_{m,d}^{p,q}({(t-x)}^4;x)\hfill \\ \hfill & =L_{m,d}^{p,q}(t^4;x)-4x{L}_{m,d}^{p,q}(t^3;x)+6x^2{L}_{m,d}^{p,q}(t^2;x)-4x^3{L}_{m,d}^{p,q}(t;x)+x^4\hfill \\ \hfill & =\frac{p^{3(m+d-1)}}{{[m]}_{p,q}^3}x-\frac{4p^{2(m+d-1)}}{{[m]}_{p,q}^2}{x}^2+\frac{3p^{2(m+d-1)}q+3p^{2m+2d-3}{q}^2+p^{m+d-2}{q}^3}{{[m]}_{p,q}^2{[m+d]}_{p,q}}{[m+d-1]}_{p,q}{x}^2\hfill \\ \hfill & +\frac{6p^{(m+d-1)}}{{[m]}_{p,q}}{x}^3-\frac{(8p^{m+d-1}q+4p^{m+d-2}{q}^2)}{{[m]}_{p,q}{[m+d]}_{p,q}}{[m+d-1]}_{p,q}{x}^3\hfill \\ \hfill & +\frac{3p^{m+d-1}{q}^3+2p^{m+d-2}{q}^4+p^{m+d-3}{q}^5}{{[m]}_{p,q}{[m+d]}_{p,q}^2}{[m+d-1]}_{p,q}{[m+d-2]}_{p,q}{x}^3\hfill \\ \hfill & +\frac{6q{[m+d-1]}_{p,q}}{{[m+d]}_{p,q}}{x}^4-\frac{4q^3}{{[m+d]}_{p,q}^2}{[m+d-1]}_{p,q}{[m+d-2]}_{p,q}{x}^4\hfill \\ \hfill & +\frac{q^6}{{[m+d]}_{p,q}^3}{[m+d-1]}_{p,q}{[m+d-2]}_{p,q}{[m+d-3]}_{p,q}{x}^4.\hfill \end{array}$$

3. Statistical approximation

First, we recall the concept of statistical convergence for sequences of real numbers which were introduced by Fast (1951) and further studied by many others. Let $K\subseteq \mathbb{N}$ and $K_n=\{j\le n:j\in K\}$. The natural density of K is defined by $\mathit{\delta }(K)={lim}_{n\to \infty }\frac{1}{n}|K_n|$ if the limit exists, where $|K_n|$ denotes the cardinality of the set $K_n$. A sequence $x=(x_k)$ of real numbers is said to be statistically convergent to L, provided that for every $\mathit{\epsilon }>0$, the set $\{j\in \mathbb{N}:|x_j-L|\ge \mathit{\epsilon }\}$ has natural density zero, that is for each $\mathit{\epsilon }>0,$$$\begin{array}{c}\hfill \underset{n}{lim}\frac{1}{n}|\{j\le n:|x_j-L|\ge \mathit{\epsilon }\}|=0.\end{array}$$

In this case, we write $st-{lim}_n{x}_n=L.$ Note that every convergent sequence is statistically convergent but not conversely. For example, let $u=(u_m)$ be defined by$$\begin{array}{c}\hfill u_m=\left\{\begin{array}{c}1\text{if}k\text{is}\text{a}\text{square},\hfill \\ 0\text{otherwise}.\hfill \end{array}\right.\end{array}$$

then, $st-lim{u}_m=0,$ but u is not convergent. Recently, the idea of statistical convergence has been used in proving some approximation theorems by various authors and it was found that the statistical versions are stronger than the classical ones. Authors have used many types of classical operators and test functions to study the Korovkin-type approximation theorems which further motivate continuation of this study. After the paper of Gadjiev and Orhan (2002), different types of summability methods have been deployed in approximation process, for example, Mursaleen, Khan, Srivastava, and Nisar (2013), Mursaleen and Kilicman (2013). In this section, we obtain the Korovkin-type weighted statistical approximation properties for these operators.

Let $C_B[0,d+1]$ be the space of all bounded and continuous functions on $[0,d+1].$ Then, $C_B[0,d+1]$ is a normed linear space with $\Vert f\Vert ={sup}_{x\ge 0}|f(x)|$. Let $\mathit{\omega }$ denote the modulus of continuity which has the following properties:

(i)	$\mathit{\omega }$ is a non-negative increasing function on $[0,d+1],$
(ii)	$\mathit{\omega }({\mathit{\delta }}_1+{\mathit{\delta }}_2)\le \mathit{\omega }({\mathit{\delta }}_1)+\mathit{\omega }({\mathit{\delta }}_1)$,
(iii)	${lim}_{\mathit{\delta }\to \infty }\mathit{\omega }(\mathit{\delta })=0.$

Let $C_B[0,d+1]$ be the space of all real-valued functions f defined on $[0,d+1]$, satisfying the following condition:$$\begin{array}{c}\hfill |f(x)-f(y)|\le \mathit{\omega }(|x-y|),\end{array}$$

for any $x,y\in [0,d+1].$ For $q\in (0,1)$ and $p\in (q,1]$ it is obvious that ${lim}_{m\to \infty }{[m]}_{p,q}=\frac{1}{p-q}.$ In order to reach to convergent result of the operator $L_{m,d}^{p,q}$, we take a sequence $q_m\in (0,1)$, $p_m\in (q,1]$ such that(3.1) $$\begin{array}{cc}\hfill \underset{m\to }{lim}{q}_m& =1,\underset{m\to }{lim}{p}_m=1\hfill \end{array}$$(3.1) (3.2) $$\begin{array}{cc}\hfill \underset{m\to }{lim}{q}_m^m& =b,\underset{m\to }{lim}{p}_m^m=a,(0<a,b\le 1).\hfill \end{array}$$(3.2)

Theorem 3.1

Let $L_{m,d}^{p,q}$ be the sequence of the operators (2.1) and the sequences $q=q_m$ , $p=p_m$ satisfy (3.1) and (3.2) and ${lim}_{m\to \infty }{[m]}_{p_m,q_m}=\infty$. Then, for any function $f\in C_B[0,d+1]$,$$\begin{array}{c}\hfill st-\underset{m\to \infty }{lim}\Vert L_{m,d}^{p_m,q_m}(f;·)-f\Vert =0.\end{array}$$

Proof

Let $e_j=t^j$, where $j=0,1,2$. Since $L_{m,d}^{p_m,q_m}(1;x)=p_m{q}_m,$ we can write$$\begin{array}{c}\hfill st-\underset{m\to \infty }{lim}\Vert L_{m,d}^{p_m,q_m}(1;x)-1\Vert =st-\underset{m\to \infty }{lim}\Vert e_0\Vert |p_m{q}_m-1|\end{array}$$

as$$\begin{array}{c}\hfill \Vert L_{m,d}^{p_m,q_m}(1;x)-1\Vert \le \Vert e_0\Vert |p_m{q}_m-1|\le |p_m{q}_m-1|,\end{array}$$

By condition (3.1), it can be observed that$$\begin{array}{c}\hfill st-\underset{m\to \infty }{lim}\Vert L_{m,d}^{p_m,q_m}(1;x)-1\Vert =0.\end{array}$$

Similarly, since $L_{m,d}^{p_m,q_m}(t;x)=p_m{q}_{m}x,$ we can write$$\begin{array}{c}\hfill st-\underset{m\to \infty }{lim}\Vert L_{m,d}^{p_m,q_m}(t;x)-x\Vert =st-\underset{m\to \infty }{lim}\Vert e_1\Vert |p_m{q}_m-1|\end{array}$$

as$$\begin{array}{c}\hfill \Vert L_{m,d}^{p_m,q_m}(t;x)-x\Vert \le \Vert e_1\Vert |p_m{q}_m-1|\le |p_m{q}_m-1|,\end{array}$$

By condition (3.1), it can be observed that$$\begin{array}{c}\hfill st-\underset{m\to \infty }{lim}\Vert L_{m,d}^{p_m,q_m}(t;x)-x\Vert =0.\end{array}$$

Lastly, we have$$\begin{array}{cc}\hfill \Vert L_{m,d}^{p_m,q_m}(t^2;x)-x^2\Vert & =\Vert e_1\Vert |\frac{p_m^{m+d-1}}{{[m]}_{p_m,q_m}}|+\Vert e_2\Vert |\frac{q_m{[m+d-1]}_{p_m,q_m}}{{[m+d]}_{p_m,q_m}}-1|\hfill \\ \hfill & \le |\frac{p_m^{m+d-1}}{{[m]}_{p_m,q_m}}|.\hfill \end{array}$$

Now for a given $\mathit{\epsilon }>0$, let us define the following sets$$\begin{array}{cc}\hfill U& =\{k:\Vert L_{m,d}^{p_m,q_m}(t^2;x)-x^2\Vert \ge \mathit{\epsilon }\},\hfill \\ \hfill U_1& =\{k:\frac{p_k^{m+d-1}}{{[m]}_{p_k,q_k}}\ge \mathit{\epsilon }\},\hfill \end{array}$$

It is obvious that $U\subseteq U_1$. Then, we obtain $\mathit{\delta }\{k\le m:\Vert L_{m,d}^{p_m,q_m}(t^2;x)-x^2\Vert \ge \mathit{\epsilon }\}$$$\begin{array}{c}\hfill \le \mathit{\delta }\{k\le m:\frac{p_k^{m+d-1}}{{[m]}_{p_k,q_k}}\ge \mathit{\epsilon }\}.\end{array}$$

By conditions (3.1) and (3.2), we have$$\begin{array}{c}\hfill st-\underset{m\to \infty }{lim}\frac{p_m^{m+d-1}}{{[m]}_{p_m,q_m}}=0\end{array}$$

So we have$$\begin{array}{c}\hfill st-\underset{m\to \infty }{lim}\Vert L_{m,d}^{p_m,q_m}(t^2;x)-x^2\Vert =0.\end{array}$$

Since$$\begin{array}{c}\hfill \Vert L_{m,d}^{p_m,q_m}(f;x)-f\Vert \le \Vert L_{m,d}^{p_m,q_m}(t^2;x)-x^2\Vert +\Vert L_{m,d}^{p_m,q_m}(t;x)-x\Vert +\Vert L_{m,d}^{p_m,q_m}(1;x)-1\Vert ,\end{array}$$

we get$$\begin{array}{cc}\hfill st-\underset{m\to \infty }{lim}\Vert L_{m,d}^{p_m,q_m}(f;x)-f\Vert & \le st-\underset{m\to \infty }{lim}\Vert L_{m,d}^{p_m,q_m}(t^2;x)-x^2\Vert +st-\underset{m\to \infty }{lim}\Vert L_{m,d}^{p_m,q_m}(t;x)-x\Vert \hfill \\ \hfill & +st-\underset{m\to \infty }{lim}\Vert L_{m,d}^{p_m,q_m}(1;x)-1\Vert ,\hfill \end{array}$$

which implies that$$\begin{array}{c}\hfill st-\underset{m\to \infty }{lim}\Vert L_{m,d}^{p_m,q_m}(f;x)-f\Vert =0.\end{array}$$

This completes the proof of the theorem. $\square$

4. Rates of convergence

We will estimate the rate of convergence in terms of modulus of continuity. Let $f\in C[0,b]$, and the modulus of continuity of f denoted by $\mathit{\omega }(f,\mathit{\delta })$ gives the maximum oscillation of f in any interval of length not exceeding $\mathit{\delta }>0$ and it is given by the relation$$\begin{array}{c}\hfill \mathit{\omega }(f,\mathit{\delta })=\underset{|y-x|\le \mathit{\delta }}{sup}|f(y)-f(x)|,x,y\in [0,b].\end{array}$$

It is known that ${lim}_{\mathit{\delta }\to 0+}\mathit{\omega }(f,\mathit{\delta })=0$ for $f\in C[0,b]$ and for any $\mathit{\delta }>0$, one has(4.1) $$\begin{array}{c}\hfill |f(y)-f(x)|\le \left(\frac{|y-x|}{\mathit{\delta }}+1\right)\mathit{\omega }(f,\mathit{\delta }).\end{array}$$(4.1)

Theorem 4.1

If $f\in C[0,d+1]$, then$$\begin{array}{c}\hfill |L_{m,d}^{p,q}(f;x)-f(x)|\le 2{\mathit{\omega }}_f({\mathit{\delta }}_m),\end{array}$$

where$$\begin{array}{c}\hfill {\mathit{\delta }}_m=\sqrt{\frac{p^{m+d-1}}{{[m]}_{p,q}}\left(1-\frac{{[m]}_{p,q}x}{{[m+d]}_{p,q}}\right)}.\end{array}$$

Proof

$$\begin{array}{cc}\hfill |L_{m,d}^{p,q}(f;x)-f(x)|& \le \frac{1}{p^{\frac{(m+d)(m+d-1)}{2}}}\sum _{k=0}^{m+d}{\left[\begin{array}{c}m+d\\ k\end{array}\right]}_{p,q}{\left(\frac{{[m]}_{p,q}x}{{[m+d]}_{p,q}}\right)}_{p,q}^k\prod _{s=0}^{m+d-k-1}\left(p^s-q^s\frac{{[m]}_{p,q}x}{{[m+d]}_{p,q}}\right)\hfill \\ & \left|f\left(\frac{{[k]}_{p,q}}{p^{k-m-d}{[m]}_{p,q}}\right)-f(x)\right|\hfill \\ & \le \frac{1}{p^{\frac{(m+d)(m+d-1)}{2}}}\sum _{k=0}^{m+d}{\left[\begin{array}{c}m+d\\ k\end{array}\right]}_{p,q}{\left(\frac{{[m]}_{p,q}x}{{[m+d]}_{p,q}}\right)}_{p,q}^k\prod _{s=0}^{m+d-k-1}\left(p^s-q^s\frac{{[m]}_{p,q}x}{{[m+d]}_{p,q}}\right)\hfill \\ & \left(\frac{\left|\frac{{[k]}_{p,q}}{p^{k-m-d}{[m]}_{p,q}}-x\right|}{\mathit{\delta }}+1\right)\mathit{\omega }(f,\mathit{\delta }).\hfill \end{array}$$

Using the Cauchy inequality and lemma (2.1), we have$$\begin{array}{cc}& |L_{m,d}^{p,q}(f;x)-f(x)|\hfill \\ & \le 1\hfill \\ & +\frac{1}{\mathit{\delta }}{\left\{\frac{1}{p^{\frac{(m+d)(m+d-1)}{2}}}\sum _{k=0}^{m+d}{\left[\begin{array}{c}m+d\\ k\end{array}\right]}_{p,q}{\left(\frac{{[m]}_{p,q}x}{{[m+d]}_{p,q}}\right)}_{p,q}^k{\left(\frac{{[k]}_{p,q}}{{[m]}_{p,q}{p}^{k-m-d}}-x\right)}^2\prod _{s=0}^{m+d-k-1}\left(p^s-q^s\frac{{[m]}_{p,q}x}{{[m+d]}_{p,q}}\right)\right\}}^{\frac{1}{2}}\hfill \\ & \left({\left(L_{m,d}^{p,q}(e_0;x)\right)}^{\frac{1}{2}}\mathit{\omega }(f,\mathit{\delta })\right)\hfill \\ & =\left\{\frac{1}{\mathit{\delta }}{\left(L_{m,d}^{p,q}(e_2;x)-2x{L}_{m,d}^{p,q}(e_1;x)+x^2{L}_{m,d}^{p,q}(e_0;x)\right)}^{\frac{1}{2}}+1\right\}\mathit{\omega }(f,\mathit{\delta })\hfill \\ & =\left\{\frac{1}{\mathit{\delta }}{\left(\frac{p^{m+d-1}}{{[m]}_{p,q}}x\left(1-\frac{{[m]}_{p,q}x}{{[m+d]}_{p,q}}\right)\right)}^{\frac{1}{2}}+1\right\}\mathit{\omega }(f,\mathit{\delta })\hfill \\ & \le \left\{\frac{1}{\mathit{\delta }}{\left(\frac{p^{m+d-1}}{{[m]}_{p,q}}\left(1-\frac{{[m]}_{p,q}x}{{[m+d]}_{p,q}}\right)\right)}^{\frac{1}{2}}+1\right\}\mathit{\omega }(f,\mathit{\delta }).\hfill \end{array}$$

Choosing$$\begin{array}{c}\hfill \mathit{\delta }={\mathit{\delta }}_m=\sqrt{\frac{p^{m+d-1}}{{[m]}_{p,q}}\left(1-\frac{{[m]}_{p,q}x}{{[m+d]}_{p,q}}\right)}.\end{array}$$

as ${lim}_{\mathit{\delta }\to 0+}$ when $m\to \infty$, we obtain the desired result. $\square$

The Peetre’s K-functional is defined by$$\begin{array}{c}\hfill K_2(f,\mathit{\delta })=inf\left\{\left(\Vert f-g\Vert +\mathit{\delta }\Vert g^{″}\Vert \right):g\in W^2\right\},\end{array}$$

where$$\begin{array}{c}\hfill W^2=\left\{g,g^{\prime },g^{″}\in C[0,d+1]\right\}.\end{array}$$

Then, there exists a positive constant $C>0$ such that $K_2(f,\mathit{\delta })\le C{\mathit{\omega }}_2\left(f,{\mathit{\delta }}^{\frac{1}{2}}\right),\mathit{\delta }>0$, where the second-order modulus of continuity is given by$$\begin{array}{c}\hfill {\mathit{\omega }}_2(f,{\mathit{\delta }}^{\frac{1}{2}})=\underset{0<h<{\mathit{\delta }}^{\frac{1}{2}}}{sup}\underset{x\in [0,d+1]}{sup}|f(x+2h)-2f(x+h)+f(x)|.\end{array}$$

Theorem 4.2

Let $f\in C[0,d+1],g^{\prime }\in C[0,d+1]$ and $0<q<p\le 1$. Then, for all $n\in \mathbb{N}$, there exists a constant $C>0$ such that$$\begin{array}{c}\hfill \left|L_{m,d}^{p,q}(f;x)-f(x)-x{g}^{\prime }(x)\left(1-\frac{{[m]}_{p,q}}{{[m+d]}_{p,q}}\right)\right|\le C{\mathit{\omega }}_2(f,{\mathit{\delta }}_m(x)),\end{array}$$

where$$\begin{array}{c}\hfill {\mathit{\delta }}_m^2(x)=\frac{p^{m+d-1}}{{[m]}_{p,q}}\left(1-\frac{{[m]}_{p,q}x}{{[m+d]}_{p,q}}\right)\end{array}$$

Proof

Let $g\in W^2$. Then, from Taylor’s expansion, we get$$\begin{array}{c}\hfill g(t)=g(x)+g^{\prime }(x)(t-x)+\underset{x}{\overset{t}{\int }}(t-u)g^{″}(u)\text{d}u,t\in [0,A],A>0.\end{array}$$

Now by lemma (2.2), we have$$\begin{array}{c}\hfill L_{m,d}^{p,q}(g;x)=g(x)+x{g}^{\prime }(x)\left(1-\frac{{[m]}_{p,q}}{{[m+d]}_{p,q}}\right)+L_{m,d}^{p,q}\left(\underset{x}{\overset{t}{\int }}(e_1-u)g^{″}(u)\text{d}u;p,q;x\right)\end{array}$$$$\begin{array}{cc}\hfill \left|L_{m,d}^{p,q}(g;x)-g(x)-x{g}^{\prime }(x)\left(1-\frac{{[m]}_{p,q}}{{[m+p]}_{p,q}}\right)\right|& \le L_{m,d}^{p,q}\left(\left|\underset{x}{\overset{t}{\int }}|(t-u)||g^{″}(u)|\text{d}u;p,q;x\right|\right)\hfill \\ \hfill & \le L_{m,d}^{p,q}\left({(t-x)}^2;p,q;x\right)\Vert g^{″}\Vert \hfill \end{array}$$

Hence, we get$$\begin{array}{c}\hfill \left|L_{m,d}^{p,q}(g;x)-g(x)-x{g}^{\prime }(x)\left(1-\frac{{[m]}_{p,q}}{{[m+d]}_{p,q}}\right)\right|\le \Vert g^{″}\Vert \left(\frac{p^{m+d-1}}{{[m]}_{p,q}}\left(1-\frac{{[m]}_{p,q}x}{{[m+d]}_{p,q}}\right)\right).\end{array}$$

On the other hand, we have$$\begin{array}{cc}\hfill \left|L_{m,d}^{p,q}(f;x)-f(x)-x{g}^{\prime }(x)\left(1-\frac{{[m]}_{p,q}}{{[m+d]}_{p,q}}\right)\right|& \le |L_{m,d}^{p,q}\left((f-g);x\right)-(f-g)(x)|\hfill \\ \hfill & +\left|L_{m,d}^{p,q}(g;x)-g(x)-x{g}^{\prime }(x)\left(1-\frac{{[m]}_{p,q}}{{[m+d]}_{p,q}}\right)\right|.\hfill \end{array}$$

Since$$\begin{array}{c}\hfill |L_{m,d}^{p,q}(f;x)|\le \Vert f\Vert ,\end{array}$$

we have$$\begin{array}{c}\hfill \left|L_{m,d}^{p,q}(f;x)-f(x)-x{g}^{\prime }(x)\left(1-\frac{{[m]}_{p,q}}{{[m+d]}_{p,q}}\right)\right|\le \Vert f-g\Vert +\Vert g^{″}\Vert \left(\frac{p^{m+d-1}}{{[m]}_{p,q}}\left(1-\frac{{[m]}_{p,q}x}{{[m+d]}_{p,q}}\right)\right).\end{array}$$

Now taking the infimum on the right-hand side over all $g\in W^2$, we get$$\begin{array}{c}\hfill \left|L_{m,d}^{p,q}(f;x)-f(x)-x{g}^{\prime }(x)\left(1-\frac{{[m]}_{p,q}}{{[m+d]}_{p,q}}\right)\right|\le C{K}_2\left(f,{\mathit{\delta }}_m^2(x)\right).\end{array}$$

In the view of the property of K-functional, we get$$\begin{array}{c}\hfill \left|L_{m,d}^{p,q}(f;x)-f(x)-x{g}^{\prime }(x)\left(1-\frac{{[m]}_{p,q}}{{[m+d]}_{p,q}}\right)\right|\le C{\mathit{\omega }}_2\left(f,{\mathit{\delta }}_m(x)\right).\end{array}$$

This completes the proof.

Now we give the rate of convergence of the operators $L_{m,d}^{p_m,q_m}(f;x)$ in terms of the elements of the usual Lipschitz class $Li{p}_M(\mathit{\gamma })$.

Let $f\in C[0,m+d]$, $M>0$ and $0<\mathit{\gamma }\le 1$. We recall that f belongs to the class $Li{p}_M(\mathit{\gamma })$ if the inequality$$\begin{array}{c}\hfill |f(t)-f(x)|\le {M|t-x|}^{\mathit{\gamma }}(t,x\in (0,1])\end{array}$$

is satisfied.

Theorem 4.3

Let $0<q<p\le 1$. Then, for each $f\in Li{p}_M(\mathit{\gamma })$, we have$$\begin{array}{c}\hfill |L_{m,d}^{p,q}(f;x)-f(x)|\le M{\mathit{\delta }}_m^{\mathit{\gamma }}(x)\end{array}$$

where$$\begin{array}{c}\hfill {\mathit{\delta }}_m^2(x)=\frac{p^{m+d-1}}{{[m]}_{p,q}}\left(1-\frac{{[m]}_{p,q}x}{{[m+d]}_{p,q}}\right)\end{array}$$

Proof

By the monotonicity of the operators $L_{m,d}^{p,q}(f;x)$, we can write$$\begin{array}{ccc}& |L_{m,d}^{p,q}(f;x)-f(x)|\hfill & \hfill \le {L^{\ast }}_{m,p,q}\left(|f(t)-f(x)|;p,q;x\right)\\ & \le \frac{1}{p^{\frac{(m+d)(m+d-1)}{2}}}\sum _{k=0}^{m+d}{\left[\begin{array}{c}m+d\\ k\end{array}\right]}_{p,q}{(r_{m,d}(p,q;x))}_{p,q}^k\prod _{s=0}^{m+d-k-1}\left(p^s-q^s{r}_{m,d}(p,q;x)\right)\hfill \\ & \left|f\left(\frac{{[k]}_{p,q}}{p^{k-m-d}{[m]}_{p,q}}\right)-f(x)\right|\hfill \\ & \le M\frac{1}{p^{\frac{(m+d)(m+d-1)}{2}}}\sum _{k=0}^{m+d}{\left[\begin{array}{c}m+d\\ k\end{array}\right]}_{p,q}{\left(\frac{{[m]}_{p,q}x}{{[m+d]}_{p,q}}\right)}_{p,q}^k\prod _{s=0}^{m+d-k-1}\left(p^s-q^s\frac{{[m]}_{p,q}x}{{[m+d]}_{p,q}}\right)\hfill \\ & {\left|\frac{{[k]}_{p,q}}{p^{k-m-d}{[m]}_{p,q}}-x\right|}^{\mathit{\gamma }}\hfill \\ & =M\sum _{k=0}^{m+d}{\left(\frac{1}{p^{\frac{(m+d)(m+d-1)}{2}}}{\mathcal{P}}_{m,d,k}(x){\left(\frac{{[k]}_{p,q}}{p^{k-m-d}{[m]}_{p,q}}-x\right)}^2\right)}^{\frac{\mathit{\gamma }}{2}}{\left(\frac{1}{p^{\frac{(m+d)(m+d-1)}{2}}}{\mathcal{P}}_{m,d,k}(x)\right)}^{\frac{2-\mathit{\gamma }}{2}},\hfill \end{array}$$

where ${\mathcal{P}}_{m,d,k}(x)={\left[\begin{array}{c}m+d\\ k\end{array}\right]}_{p,q}{\left(\frac{{[m]}_{p,q}x}{{[m+d]}_{p,q}}\right)}_{p,q}^k\prod _{s=0}^{m+d-k-1}\left(p^s-q^s\frac{{[m]}_{p,q}x}{{[m+d]}_{p,q}}\right)$

Now applying the Hölder’s inequality$$\begin{array}{cc}& |L_{m,d}^{p,q}(f;x)-f(x)|\hfill \\ \hfill & \le M{\left(\frac{1}{p^{\frac{(m+d)(m+d-1)}{2}}}\sum _{k=0}^{m+d}{\mathcal{P}}_{m,d,k}(x){\left(\frac{{[k]}_{p,q}}{p^{k-m-d}{[m]}_{p,q}}-x\right)}^2\right)}^{\frac{\mathit{\gamma }}{2}}\hfill \\ \hfill & {\left(\frac{1}{p^{\frac{(m+d)(m+d-1)}{2}}}\sum _{k=0}^{m+d}{\mathcal{P}}_{m,d,k}(x)\right)}^{\frac{2-\mathit{\gamma }}{2}}\hfill \\ \hfill & =M{\left(L_{m,d}^{p,q}\left({(t-x)}^2;x\right)\right)}^{\frac{\mathit{\gamma }}{2}}\hfill \end{array}$$

Choosing $\mathit{\delta }:{\mathit{\delta }}_m(x)=\sqrt{L_{m,d}^{p,q}\left({(t-x)}^2;x\right)}$,

we obtain$$\begin{array}{c}\hfill |L_{m,d}^{p,q}(f;x)-f(x)|\le M{\mathit{\delta }}_m^{\mathit{\gamma }}(x).\end{array}$$

Hence, the desired result is obtained. $\square$

5. Voronovskaja-type theorem

Theorem 5.1

Let $f\in C[0,d+1]$ be such that $f^{\prime },f^{″}\in C[0,d+1]$. Let the sequences $\{p_m\}$, $\{q_m\}$ satisfy $0<q_m<p_m\le 1$ such that $p_m\to 1,q_m\to 1$ and $p_m^m\to a,q_m^m\to b$ as $m\to \infty$, where $0\le a,b<1$. Suppose that ${lim}_{m\to \infty }{[m]}_{p_m,q_m}=\infty$. Then$$\begin{array}{c}\hfill \underset{m\to \infty }{lim}{[m]}_{p_m,q_m}\left(L_{m,d}^{p_m,q_m}(f;x)-f(x)\right)=\frac{x(\mathit{\lambda }-ax)}{2}{f}^{″}(x),\end{array}$$

uniformly on $[0,d+1]$, where $0<\mathit{\lambda }\le 1.$

Proof

By Taylor’s formula, we may write$$\begin{array}{c}\hfill f(t)=f(x)+f^{\prime }(x)(t-x)+\frac{1}{2}{f}^{″}(x){(t-x)}^2+r(t,x){(t-x)}^2\end{array}$$

where r(t, x) is the remainder term and ${lim}_{t\to x}r(t,x)=0$. Therefore, we have$$\begin{array}{cc}& {[m]}_{p_m,q_m}\left(L_{m,d}^{p_m,q_m}(f;x)-f(x)\right)\hfill \\ \hfill & ={[m]}_{p_m,q_m}\left(f^{\prime }(x)L_{m,d}^{p_m,q_m}\left((t-x);x\right)+\frac{f^{″}(x)}{2}{L}_{m,d}^{p_m,q_m}\left({(t-x)}^2;x\right)+L_{m,d}^{p_m,q_m}(r(t,x){(t-x)}^2;x)\right).\hfill \end{array}$$

By the Cauchy–Schwartz inequality, we have(5.1) $$\begin{array}{c}\hfill L_{m,d}^{p_m,q_m}\left(r(t,x){(t-x)}^2;x)\right)\le \sqrt{L_{m,d}^{p_m,q_m}\left(r^2(t,x);x)\right)}·\sqrt{L_{m,d}^{p_m,q_m}\left({(t-x)}^4;x)\right)}.\end{array}$$(5.1)

Observe that $r^2(x,x)=0$, and $r^2(t,x)\in C[0,d+1]$; then, it follows from Theorem 3.1 that(5.2) $$\begin{array}{c}\hfill L_{m,d}^{p_m,q_m}\left(r^2(t,x);x)\right)=r^2(x,x)=0,\end{array}$$(5.2)

uniformly with respect to $x\in C[0,d+1]$, in view of the fact that $L_{m,d}^{p_m,q_m}\left({(t-x)}^4;x)\right)=o\left(\frac{1}{{[m]}_{p_m,q_m}^2}\right)$. Now from (5.1), (5.2) and Lemma 2.2 (ii), we get(5.3) $$\begin{array}{c}\hfill L_{m,d}^{p_m,q_m}\left(r(t,x){(t-x)}^2;x)\right)=0\end{array}$$(5.3)

Now we compute the following:(5.4) $$\begin{array}{cc}& \underset{m\to \infty }{lim}{[m]}_{p_m,q_m}\left(L_{m,d}^{p_m,q_m}\left((t-x);x)\right)\right)=0,\hfill \\ \hfill & \underset{m\to \infty }{lim}{[m]}_{p_m,q_m}{L}_{m,d}^{p_m,q_m}\left({(t-x)}^2;x)\right)\hfill \\ \hfill & =x\underset{m\to \infty }{lim}{[m]}_{p_m,q_m}\frac{1}{{[m]}_{p_m,q_m}}{p}_m^{m+d-1}\left(1-\frac{{[m]}_{p,q}x}{{[m+d]}_{p,q}}\right)\hfill \\ \hfill & \underset{m\to \infty }{lim}{[m]}_{p_m,q_m}\left(L_{m,d}^{p_m,q_m}\left({(t-x)}^2;x)\right)\right)=\mathit{\lambda }x-a{x}^2=x(\mathit{\lambda }-ax).\hfill \end{array}$$(5.4)

where $\mathit{\lambda }\in (0,1]$ depending on the sequence $\{p_m\}$.

Finally, from (5.3), (5.4) and (5.5), we get the required result. This completes the proof of the theorem. $\square$

Additional information

Funding

The authors received no direct funding for this research.

Notes on contributors

M. Mursaleen

The first author is the PhD supervisor of other two co-authors. Presently, we have two groups of students working on different topics, e.g. sequence spaces, measures of non-compactness, approximation theory, differential and integral equations. One of the groups is working on approximation of positive linear operators, their q- and (p, q)-generalizations. Presently, the first author is a full-professor and chairman of the Department of Mathematics. Recently, he has received the award of Outstanding Researcher of the Year-2014 of Aligarh Muslim University.