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Approximation by \((p,q)\)-Lupaş–Schurer–Kantorovich operators

Abstract

In the current paper, we examine the \((p,q)\)-analogue of Kantorovich type Lupaş–Schurer operators with the help of \((p,q)\)-Jackson integral. Then, we estimate the rate of convergence for the constructed operators by using the modulus of continuity in terms of a Lipschitz class function and by means of Peetre’s K-functionals based on Korovkin theorem. Moreover, we illustrate the approximation of the \((p,q)\)-Lupaş–Schurer–Kantorovich operators to appointed functions by the help of Matlab algorithm and then show the comparison of the convergence of these operators with Lupaş–Schurer operators based on \((p,q)\)-integers.

Introduction

In 1962, Bernstein–Schurer operators were identified in the paper of Schurer [25]. In 1987, Lupaş [16] initiated the q-generalization of Bernstein operators in rational form. Some other q-Bernstein polynomial was defined by Phillips [22] in 1997. The development q-calculus applications established a precedent in the field of approximation theory. We may refer to some of them as Durrmeyer variant of q-Bernstein–Schurer operators [2], q-Bernstein–Schurer–Kantorovich type operators [3], q-Durrmeyer operators [8], q-Bernstein–Schurer–Durrmeyer type operators [12], q-Bernstein–Schurer operators [19], King’s type modified q-Bernstein–Kantorovich operators [20], q-Bernstein–Schurer–Kantorovich operators [23]. Lately, Mursaleen et al. [17] pioneered the research of \((p,q)\)-analogue of Bernstein operators which is a generalization of q-Bernstein operators (Philips). The application of \((p,q)\)-calculus has led to the discovery of various modifications of Bernstein polynomials involving \((p,q)\)-integers. For instance, Mursaleen et al. [18] constructed \((p,q)\)-analogue of Bernstein-Kantorovich operators in 2016, and Khalid et al. [15] generalised q-Bernstein–Lupaş operators. In the \((p,q)\)-calculus, parameter p provides suppleness to the approximation. Some recent articles are [1, 46, 9, 10, 13], and [21]. Motivated by the work of Khalid et al. [15], now we define a Kantorovich type Lupaş-Schurer operators based on the \((p,q)\)-calculus.

First of all, we introduce some important notations and definitions for the \((p,q)\)-calculus, which is a generalization of q-oscillator algebras. For \(0< q< p\leq1\) and \(m\geq0\), the \((p,q)\)-number of m is denoted by \([m]_{p,q}\) and is defined by

$$ [m]_{p,q}:=p^{m-1}+p^{m-2}q+\cdots+pq^{m-2}+q^{m-1}= \textstyle\begin{cases} \frac{p^{m}-q^{m}}{p-q} & \text{if } p\neq q\neq1, \\ \frac{1-q^{m}}{1-q} & \text{if } p= 1, \\ m & \text{if } p=q=1. \end{cases} $$

The formula for the \((p,q)\)-binomial expansion is defined by

( c x + d y ) p , q m := l = 0 m [ m l ] p , q p ( m l ) ( m l 1 ) 2 q l ( l 1 ) 2 c m l d l x m l y l ,
(1)

where

[ m l ] p , q = [ m ] p , q ! [ l ] p , q ! [ m l ] p , q !

are the \((p,q)\)-binomial coefficients. From Eq. (1) we get

$$ (x+y)_{p,q}^{m} =(x+y) (px+qy) \bigl(p^{2}x+q^{2}y \bigr)\cdots \bigl(p^{m-1}x+q^{m-1}y \bigr) $$

and

$$ (1-x)_{p,q}^{m} =(1-x) (p-qx) \bigl(p^{2}-q^{2}x \bigr)\cdots \bigl(p^{m-1}-q^{m-1}x \bigr). $$

The \((p,q)\)-Jackson integrals are defined by

$$ \int_{0}^{a}f(x)\,d_{p,q}x=(q-p)a \sum _{k=0}^{ \infty} \frac{p^{k}}{q^{k+1}}f \biggl( \frac{p^{k}}{q^{k+1}}a \biggr), \quad \biggl\vert \frac{p}{q} \biggr\vert < 1 $$

and

$$ \int_{0}^{a}f(x) \,d_{p,q}x=(p-q)a \sum _{k=0}^{ \infty} \frac{q^{k}}{p^{k+1}}F \biggl( \frac{q^{k}}{p^{k+1}}a \biggr),\quad \biggl\vert \frac{q}{p} \biggr\vert < 1. $$

For detailed information about the theory of \((p,q)\)-integers, we refer to [11] and [24].

Construction of the operator

Definition 1

For any \(0< q< p\leq1\), we construct a \((p,q)\)-analogue of Kantorovich type Lupaş–Schurer operator by

$$ K_{m,s}^{(p,q)}(f;x)=[m]_{p,q}\sum _{l=0}^{m+s}\frac{B_{m,l,s}^{p,q}(x)}{p^{m-l}q^{l}} \int_{\frac{[l]_{p,q}}{p^{l-m-1}[m]_{p,q}}}^{\frac {[l+1]_{p,q}}{p^{l-m}[m]_{p,q}}} f(t)\,d_{p,q}t,\quad x \in [0,1], $$
(2)

where \(m\in\mathbb{N}\), \(f\in C[0,s+1]\), \(s>0\) is a fixed natural number and

B m , l , s p , q (x)= [ m + s l ] p , q p ( m + s l ) ( m + s l 1 ) 2 q l ( l 1 ) 2 x l ( 1 x ) m + s l j = 1 m + s { p j 1 ( 1 x ) + q j 1 x } .
(3)

After some calculations we obtain

$$ K_{m,s}^{(p,q)}(f;x)=\sum _{l=0}^{m+s}B_{m,l,s}^{p,q}(x) \int_{0}^{1}f \biggl( \frac{p[l]_{p,q}+q^{l}t}{p^{l-m}[m]_{p,q}} \biggr) \,d_{p,q}t. $$
(4)

In the following lemma, we present some equalities for the \((p,q)\)-analogue of Lupaş–Schurer–Kantorovich operators.

Lemma 1

Let \(K_{m,s}^{(p,q)}(\cdot;\cdot)\) be given by Eq. (4). Then we have

$$\begin{aligned}& K_{m,s}^{(p,q)}(1;x) = 1, \end{aligned}$$
(5)
$$\begin{aligned}& K_{m,s}^{(p,q)}(t;x) = \biggl( \frac{[m+s]_{p,q}}{[m]_{p,q}p^{s-1}} - \frac{p^{m}}{[2]_{p,q}[m]_{p,q}}+ \frac{q^{m+s}}{[2]_{p,q}[m]_{p,q}p^{s}} \biggr)x+\frac{p^{m}}{[2]_{p,q}[m]_{p,q}}, \end{aligned}$$
(6)
$$\begin{aligned}& \begin{aligned}[b] K_{m,s}^{(p,q)} \bigl(t^{2};x \bigr) &= \frac {[m+s]_{p,q}[m+s-1]_{p,q}q^{2}p^{2-2s}}{[m]_{p,q}^{2}(p(1-x)+qx)}x^{2}+\frac {[m+s]_{p,q}p^{m-s+1}}{[m]_{p,q}^{2}}x \\ &\quad {}+\frac {2[m+s]_{p,q}qp^{4m+2s-3}(p^{m+s}(1-x)+q^{m+s}x)}{[2]_{p,q}[m]_{p,q}^{2}(p(1-x)+qx)}x \\ &\quad {}+\frac {p^{-2s}(p^{m+s}(1-x)+q^{m+s}x)(p^{m+s+1}(1-x)+q^{m+s+1}x)}{[3]_{p,q}[m]_{p,q}^{2}(p(1-x)+qx)}, \end{aligned} \end{aligned}$$
(7)
$$\begin{aligned}& K_{m,s}^{(p,q)}(t-x;x) = \biggl( \frac{[m+s]_{p,q}}{[m]_{p,q}p^{s-1}} - \frac{p^{m}}{[2]_{p,q}[m]_{p,q}}+ \frac{q^{m+s}}{[2]_{p,q}[m]_{p,q}p^{s}}-1 \biggr)x+\frac{p^{m}}{[2]_{p,q}[m]_{p,q}}, \end{aligned}$$
(8)
$$\begin{aligned}& \begin{aligned}[b] K_{m,s}^{(p,q)} \bigl((t-x)^{2};x \bigr) &= \biggl( \frac {[m+s]_{p,q}[m+s-1]_{p,q}q^{2}p^{2-2s}}{[m]_{p,q}^{2}(p(1-x)+qx)} \\ &\quad {}+\frac {-2[2]_{p,q}[m+s]_{p,q}p^{1-s}+2p^{m}-2q^{m+s}p^{-s}}{[2]_{p,q}[m]_{p,q}}+1 \biggr) x^{2} \\ &\quad {}+ \biggl(\frac{[m+s]_{p,q}p^{m-s+1}}{[m]_{p,q}^{2}} +\frac {2[m+s]_{p,q}qp^{4m+2s-3}(p^{m+s}(1-x)+q^{m+s}x)}{[2]_{p,q}[m]_{p,q}^{2}(p(1-x)+qx)}\hspace{-20pt} \\ &\quad {}-\frac{2p^{m}}{[2]_{p,q}[m]_{p,q}} \biggr)x \\ &\quad {}+ \frac {p^{-2s}(p^{m+s}(1-x)+q^{m+s}x)(p^{m+s+1}(1-x)+q^{m+s+1}x)}{[3]_{p,q}[m]_{p,q}^{2}(p(1-x)+qx)}. \end{aligned} \end{aligned}$$
(9)

Proof

(i) From the definition of the operators in (4), we can easily prove the first claim as follows:

K m , s ( p , q ) ( 1 ; x ) = l = 0 m + s B m , l , s p , q ( x ) 0 1 d p , q t = l = 0 m + s [ m + s l ] p , q p ( m + s l ) ( m + s l 1 ) 2 q l ( l 1 ) 2 x l ( 1 x ) m + s l j = 1 m + s { p j 1 ( 1 x ) + q j 1 x } = 1 .
(10)

(ii) We can calculate the second identity for \(K_{m,s}^{(p,q)}(t;x)\) as follows:

$$\begin{aligned} K_{m,s}^{(p,q)}(t;x) =&\sum_{l=0}^{m+s}B_{m,l,s}^{p,q}(x) \int_{0}^{1} \frac{p[l]_{p,q}+q^{l}t}{p^{l-m}[m]_{p,q}} \,d_{p,q}t \\ =&\sum_{l=0}^{m+s}B_{m,l,s}^{p,q}(x) \frac{p[l]_{p,q}}{p^{l-m}[m]_{p,q}} \int_{0}^{1} d_{p,q}t+\sum _{l=0}^{m+s}B_{m,l,s}^{p,q}(x) \frac{q^{l}}{p^{l-m}[m]_{p,q}} \int_{0}^{1} t \,d_{p,q}t . \end{aligned}$$

After that, by some simple computations, we have

K m , s ( p , q ) ( t ; x ) = l = 0 m + s B m , l , s p , q ( x ) p [ l ] p , q p l m [ m ] p , q + l = 0 m + s B m , l , s p , q ( x ) q l p l m [ m ] p , q [ 2 ] p , q = l = 1 m + s p m l + 1 [ m + s ] p , q [ m ] p , q . [ m + s 1 l 1 ] p , q p ( m + s l ) ( m + s l 1 ) 2 q l ( l 1 ) 2 x l ( 1 x ) m + s l j = 1 m + s { p j 1 ( 1 x ) + q j 1 x } + 1 [ m ] p , q [ 2 ] p , q p s l = 0 m + s [ m + s l ] p , q p ( m + s l ) ( m + s l 1 ) 2 q l ( l 1 ) 2 ( q x p ( 1 x ) ) l j = 0 m + s 1 { p j 1 + q j 1 ( q x p ( 1 x ) ) } = [ m + s ] p , q [ m ] p , q p s l = 0 m + s 1 p m + s l [ m + s 1 l ] p , q p ( m + s l 1 ) ( m + s l 2 ) 2 q l ( l + 1 ) 2 x l + 1 ( 1 x ) m + s l 1 j = 1 m + s 1 { p j ( 1 x ) + q j x } + p ( 1 x ) { p m + s 1 + q m + s 1 ( q x p ( 1 x ) ) } [ m ] p , q [ 2 ] p , q p s × l = 0 m + s [ m + s l ] p , q p ( m + s l ) ( m + s l 1 ) 2 q l ( l 1 ) 2 ( q x p ( 1 x ) ) l j = 1 m + s { p j 1 + q j 1 ( q x p ( 1 x ) ) } = [ m + s ] p , q [ m ] p , q p s 1 x + p ( 1 x ) { p m + s 1 + q m + s 1 ( q x p ( 1 x ) ) } [ m ] p , q [ 2 ] p , q p s .

Then, \(K_{m,s}^{(p,q)}(t;x)\) is obtained as

$$ K_{m,s}^{(p,q)}(t;x)= \biggl( \frac{[m+s]_{p,q}}{[m]_{p,q}p^{s-1}} - \frac{p^{m}}{[2]_{p,q}[m]_{p,q}}+ \frac{q^{m+s}}{[2]_{p,q}[m]_{p,q}p^{s}} \biggr)x+\frac{p^{m}}{[2]_{p,q}[m]_{p,q}}. $$

Thus, (6) is obtained.

(iii) For the third identity involving \(K_{m,s}^{(p,q)}(t^{2};x)\), we write

$$\begin{aligned} K_{m,s}^{(p,q)} \bigl(t^{2};x \bigr) =&\sum _{l=0}^{m+s}B_{m,l,s}^{p,q}(x) \frac{p^{2}[l]^{2}_{p,q}}{p^{2l-2m}[m]^{2}_{p,q}} \int_{0}^{1} d_{p,q}t+2\sum _{l=0}^{m+s}B_{m,l,s}^{p,q}(x) \frac{p[l]_{p,q}q^{l}}{p^{2l-2m}[m]^{2}_{p,q}} \int_{0}^{1} t \,d_{p,q}t \\ &{}+\sum_{l=0}^{m+s}B_{m,l,s}^{p,q}(x) \frac{q^{2l}}{p^{2l-2m}[m]^{2}_{p,q}} \int_{0}^{1} t^{2} \,d_{p,q}t \\ =& \underbrace{\sum_{l=0}^{m+s}B_{m,l,s}^{p,q}(x) \frac{p^{2}[l]^{2}_{p,q}}{p^{2l-2m}[m]^{2}_{p,q}}}_{\text{B1}} + \underbrace {\frac{2}{[2]_{p,q}}\sum _{l=0}^{m+s}B_{m,l,s}^{p,q}(x) \frac{p[l]_{p,q}q^{l}}{p^{2l-2m}[m]^{2}_{p,q}}}_{\text{B2}} \\ &{}+ \underbrace{\frac{1}{[3]_{p,q}}\sum_{l=0}^{m+s}B_{m,l,s}^{p,q}(x) \frac{q^{2l}}{p^{2l-2m}[m]^{2}_{p,q}}}_{\text{B3}}. \end{aligned}$$
(11)

Firstly, we calculate B1 as

B 1 = l = 0 m + s B m , l , s p , q ( x ) p 2 [ l ] p , q 2 p 2 l 2 m [ m ] p , q 2 = l = 0 m + s 1 p 2 m 2 l [ l + 1 ] p , q [ m + s ] p , q [ m ] p , q 2 . [ m + s 1 l ] p , q p ( m + s l 1 ) ( m + s l 2 ) 2 q l ( l + 1 ) 2 x l + 1 ( 1 x ) m + s l 1 j = 1 m + s { p j 1 ( 1 x ) + q j 1 x } .

Now by using the equality

$$ [l+1]_{p,q}=p^{l}+q[l]_{p,q}, $$
(12)

we acquire

B 1 = [ m + s ] p , q [ m ] p , q 2 l = 0 m + s 1 p 2 m l [ m + s 1 l ] p , q p ( m + s l 1 ) ( m + s l 2 ) 2 q l ( l + 1 ) 2 x l + 1 ( 1 x ) m + s l 1 j = 1 m + s { p j 1 ( 1 x ) + q j 1 x } + [ m + s ] p , q [ m ] p , q 2 l = 0 m + s 1 p 2 m 2 l q [ l ] p , q [ m + s 1 l ] p , q p ( m + s l 1 ) ( m + s l 2 ) 2 q l ( l + 1 ) 2 x l + 1 ( 1 x ) m + s l 1 j = 1 m + s { p j 1 ( 1 x ) + q j 1 x } = [ m + s ] p , q p 2 m x [ m ] p , q 2 p m + s 1 l = 0 m + s 1 [ m + s 1 l ] p , q p ( m + s l 1 ) ( m + s l 2 ) 2 q l ( l 1 ) 2 ( q x p ( 1 x ) ) l ( 1 x ) m + s 1 1 p m + s 1 j = 1 m + s 1 { p j ( 1 x ) + q j x } + [ m + s ] p , q [ m + s 1 ] p , q q 2 x 2 [ m ] p , q 2 p 2 s 2 ( p ( 1 x ) + q x ) l = 0 m + s 2 [ m + s 2 l ] p , q p ( m + s l 2 ) ( m + s l 3 ) 2 q l ( l 1 ) 2 ( q 2 x p 2 ( 1 x ) ) l j = 1 m + s 2 { p j 1 + q j 1 ( q 2 x p 2 ( 1 x ) ) } = [ m + s ] p , q p m s + 1 [ m ] p , q 2 x + [ m + s ] p , q [ m + s 1 ] p , q p 2 2 s q 2 [ m ] p , q 2 ( p ( 1 x ) + q x ) x 2 .
(13)

Secondly, we work out B2 as follows:

B 2 = 2 [ 2 ] p , q l = 0 m + s B m , l , s p , q ( x ) p [ l ] p , q q l p 2 l 2 m [ m ] p , q 2 = 2 [ m + s ] p , q [ 2 ] p , q [ m ] p , q 2 l = 1 m + s q l p 2 l 2 m 1 . [ m + s 1 l 1 ] p , q p ( m + s l ) ( m + s l 1 ) 2 q l ( l 1 ) 2 x l ( 1 x ) m + s l j = 1 m + s { p j 1 ( 1 x ) + q j 1 x } = 2 [ m + s ] p , q x [ 2 ] p , q [ m ] p , q 2 l = 0 m + s 1 q p 2 m + 1 . [ m + s 1 l ] p , q p ( m + s l 1 ) ( m + s l 2 ) 2 q l ( l 1 ) 2 ( q 2 x p 2 ( 1 x ) ) l ( 1 x ) m + s 1 j = 2 m + s { p j 1 ( 1 x ) + q j 1 x } = 2 [ m + s ] p , q q p 2 m 1 x [ 2 ] p , q [ m ] p , q 2 l = 0 m + s 1 [ m + s 1 l ] p , q p ( m + s l 1 ) ( m + s l 2 ) 2 q l ( l 1 ) 2 ( q 2 x p 2 ( 1 x ) ) l ( 1 x ) m + s 1 j = 0 m + s 2 { p j + 1 ( 1 x ) + q j + 1 x } = 2 [ m + s ] p , q q p 4 m + 2 s 3 x [ 2 ] p , q [ m ] p , q 2 l = 0 m + s 1 [ m + s 1 l ] p , q p ( m + s l 1 ) ( m + s l 2 ) 2 q l ( l 1 ) 2 ( q 2 x p 2 ( 1 x ) ) l j = 0 m + s 2 { p j 1 + q j 1 ( q 2 x p 2 ( 1 x ) ) } = 2 [ m + s ] p , q q p 4 m + 2 s 3 [ 2 ] p , q [ m ] p , q 2 . ( p m + s ( 1 x ) + q m + s x ) p ( 1 x + q x ) x .
(14)

Thirdly, we deal with B3 as

B 3 = 1 [ 3 ] p , q l = 0 m + s B m , l , s p , q ( x ) q 2 l p 2 l 2 m [ m ] p , q 2 = p 2 m [ 3 ] p , q [ m ] p , q 2 l = 0 m + s [ m + s l ] p , q p ( m + s l ) ( m + s l 1 ) 2 q l ( l 1 ) 2 ( q 2 x p 2 ( 1 x ) ) l ( 1 x ) m + s j = 0 m + s 2 { p j + 1 ( 1 x ) + q j + 1 x } = p 2 s [ 3 ] p , q [ m ] p , q 2 . ( p m + s ( 1 x ) + q m + s x ) ( p m + s + 1 ( 1 x ) + q m + s + 1 x ) p ( 1 x ) + q x .
(15)

As a consequence, \(K_{m,s}^{(p,q)}(t^{2};x)\) is found as

$$\begin{aligned} K_{m,s}^{(p,q)} \bigl(t^{2};x \bigr) =& \frac{[m+s]_{p,q}p^{m-s+1}}{[m]^{2}_{p,q}}x+\frac {[m+s]_{p,q}[m+s-1]_{p,q}p^{2-2s}q^{2}}{[m]^{2}_{p,q}(p(1-x)+qx)}x^{2} \\ &{}+\frac{2[m+s]_{p,q}qp^{4m+2s-3}}{[m]^{2}_{p,q}[2]_{p,q}}.\frac {(p^{m+s}(1-x)+q^{m+s}x)}{p(1-x+qx)}x \\ &{}+\frac{p^{-2s}}{[3]_{p,q}[m]^{2}_{p,q}}.\frac {(p^{m+s}(1-x)+q^{m+s}x)(p^{m+s+1}(1-x)+q^{m+s+1}x)}{p(1-x)+qx}. \end{aligned}$$

If we reorganize, we obtain

$$\begin{aligned} K_{m,s}^{(p,q)} \bigl(t^{2};x \bigr) =& \frac {[m+s]_{p,q}[m+s-1]_{p,q}q^{2}p^{2-2s}}{[m]_{p,q}^{2}(p(1-x)+qx)}x^{2}+\frac {[m+s]_{p,q}p^{m-s+1}}{[m]_{p,q}^{2}}x \\ &{}+\frac {2[m+s]_{p,q}qp^{4m+2s-3}(p^{m+s}(1-x)+q^{m+s}x)}{[2]_{p,q}(p(1-x)+qx)[m]_{p,q}^{2}}x \\ &{}+\frac {p^{-2s}(p^{m+s}(1-x)+q^{m+s}x)(p^{m+s+1}(1-x)+q^{m+s+1}x)}{[3]_{p,q}[m]_{p,q}^{2}(p(1-x)+qx)} , \end{aligned}$$
(16)

as desired.

(iv) By using the linearity of the operators \(K_{m,s}^{(p,q)}\), we acquire the first central moment \(K_{m,s}^{(p,q)}(t-x;x)\) as

$$\begin{aligned} K_{m,s}^{(p,q)}(t-x;x) =&K_{m,s}^{(p,q)}(t;x)-xK_{m,s}^{(p,q)}(1;x) \\ =& \biggl( \frac{[m+s]_{p,q}}{[m]_{p,q}p^{s-1}} -\frac {p^{m}}{[2]_{p,q}[m]_{p,q}}+ \frac{q^{m+s}}{[2]_{p,q}[m]_{p,q}p^{s}}-1 \biggr)x \\ &{}+\frac{p^{m}}{[2]_{p,q}[m]_{p,q}}. \end{aligned}$$
(17)

(v) Similarly, we write the second central moment \(K_{m,s}^{(p,q)}((t-x)^{2};x)\) as

$$ K_{m,s}^{(p,q)} \bigl((t-x)^{2};x \bigr)=K_{m,s}^{(p,q)} \bigl(t^{2};x \bigr)-2xK_{m,s}^{(p,q)}(t;x)+x^{2}K_{m,s}^{(p,q)}(1;x). $$
(18)

We now plug-in into equation (18) expressions (5), (6) and (7). Then we get

$$\begin{aligned} K_{m,s}^{(p,q)} \bigl((t-x)^{2};x \bigr) =& \biggl( \frac {[m+s]_{p,q}[m+s-1]_{p,q}q^{2}p^{2-2s}}{[m]_{p,q}^{2}(p(1-x)+qx)} \\ &{}+\frac {-2[2]_{p,q}[m+s]_{p,q}p^{1-s}+2p^{m}-2q^{m+s}p^{-s}}{[2]_{p,q}[m]_{p,q}}+1 \biggr) x^{2} \\ &{}+ \biggl(\frac{[m+s]_{p,q}p^{m-s+1}}{[m]_{p,q}^{2}} +\frac {2[m+s]_{p,q}qp^{4m+2s-3}(p^{m+s}(1-x)+q^{m+s}x)}{[2]_{p,q}[m]_{p,q}^{2}(p(1-x)+qx)} \\ &{}-\frac {2p^{m}}{[2]_{p,q}[m]_{p,q}} \biggr)x \\ &{}+ \frac {p^{-2s}(p^{m+s}(1-x)+q^{m+s}x)(p^{m+s+1}(1-x)+q^{m+s+1}x)}{[3]_{p,q}(p(1-x)+qx)[m]_{p,q}^{2}}. \end{aligned}$$
(19)

 □

We can easily see that \(K_{m,s}^{(p,q)}(f;x)\) are linear positive operators.

Remark 1

[15] Let p, q satisfy \(0< q< p \leq1\) and \(\lim_{m\rightarrow\infty}[m]_{p,q}=\frac{1}{p-q}\). To obtain the convergence results for operators \(K_{m,s}^{(p,q)}(f;x)\), we take sequences \(q_{m}\in(0,1)\), \(p_{m}\in (q_{m},1]\) such that \(\lim_{m\rightarrow\infty}p_{m}=1\), \(\lim_{m\rightarrow\infty}q_{m}=1\), \(\lim_{m\rightarrow\infty}p_{m}^{m}=1\) and \(\lim_{m\rightarrow\infty}q_{m}^{m}=1\). Such sequences can be constructed by taking \(p_{m}=1-1/m^{2}\) and \(q_{m}=1-1/2m^{2}\).

Now we will present the next theorem, which ensures the approximation process according to Korovkin’s approximation theorem.

Theorem 1

Let \(K_{m,s}^{(p,q)}(f;x)\) satisfy the conditions \(p_{m}\rightarrow1\), \(q_{m}\rightarrow1\), \(p_{m}^{m}\rightarrow1\) and \(q_{m}^{m}\rightarrow1\) as \(m\rightarrow\infty\) for \(q_{m}\in(0,1)\), \(p_{m}\in(q_{m},1]\). Then for every monotone increasing function \(f\in C [ 0,s+1 ]\), operators \(K_{m,s}^{(p,q)}(f;x)\) converge uniformly to f.

Proof

By the Korovkin theorem, it is sufficient to prove that

$$ \lim_{m\longrightarrow\infty} \bigl\Vert K_{m,s}^{(p,q)}e_{k}-e_{k} \bigr\Vert =0,\quad k=0,1,2, $$

where \(e_{k}(x)=x^{k}\), \(k=0,1,2\).

(i) By using Eq. (5), it can be clearly seen that

$$ \lim_{m\longrightarrow\infty} \bigl\Vert K_{m,s}^{(p,q)}e_{0}-e_{0} \bigr\Vert =\lim_{m\longrightarrow\infty}\sup_{x \in[0,1] } \bigl\vert K_{m,s}^{(p,q)}(1;x)-1 \bigr\vert =0. $$

(ii) By Eq. (6), we write

$$\begin{aligned}& \lim_{m\longrightarrow\infty} \bigl\Vert K_{m,s}^{(p,q)}e_{1}-e_{1} \bigr\Vert \\& \quad = \lim_{m\longrightarrow\infty}\sup_{x \in[0,1] } \bigl\vert K_{m,s}^{(p,q)}(t;x)-x \bigr\vert \\& \quad = \lim_{m\longrightarrow\infty}\sup_{x \in[0,1] } \biggl\vert \biggl( \frac{[m+s]_{p,q}}{p^{s-1}[m]_{p,q}} -\frac {p^{m}}{[2]_{p,q}[m]_{p,q}}+ \frac{q^{m+s}}{[2]_{p,q}[m]_{p,q}p^{s}}-1 \biggr)x +\frac{p^{m}}{[2]_{p,q}[m]_{p,q}} \biggr\vert \\& \quad \leq \lim_{m\longrightarrow\infty} \biggl( \frac {[m+s]_{p,q}}{p^{s-1}[m]_{p,q}}-1+ \frac{q^{m+s}}{[2]_{p,q}[m]_{p,q}p^{s}} \biggr) \\& \quad = 0. \end{aligned}$$

(iii) From Eq. (7), we have

$$\begin{aligned}& \lim_{m\longrightarrow\infty} \bigl\Vert K_{m,s}^{(p,q)}e_{2}-e_{2} \bigr\Vert \\& \quad = \lim_{m\longrightarrow\infty}\sup_{x \in[0,1] } \bigl\vert K_{m,s}^{(p,q)} \bigl(t^{2};x \bigr)-x^{2} \bigr\vert \\& \quad = \lim_{m\longrightarrow\infty}\sup_{x \in[0,1] } \biggl\vert \biggl( \frac {[m+s]_{p,q}[m+s-1]_{p,q}q^{2}p^{2-2s}}{[m]_{p,q}^{2}(p(1-x)+qx)}-1 \biggr) x^{2} \\& \qquad {}+\frac{[m+s]_{p,q}p^{m-s+1}}{[m]_{p,q}^{2}}x +\frac {2[m+s]_{p,q}qp^{4m+2s-3}(p^{m+s}(1-x)+q^{m+s}x)}{[2]_{p,q}[m]_{p,q}^{2}(p(1-x)+qx)}x \\& \qquad {}+\frac {p^{-2s}(p^{m+s}(1-x)+q^{m+s}x)(p^{m+s+1}(1-x)+q^{m+s+1}x)}{[3]_{p,q}[m]_{p,q}^{2}(p(1-x)+qx)} \biggr\vert \\& \quad \leq \lim_{m\longrightarrow\infty} \biggl( \biggl( \frac {[m+s]_{p,q}[m+s-1]_{p,q}q^{2}p^{2-2s}}{[m]_{p,q}^{2}(p(1-x)+qx)}-1 \biggr) +\frac{[m+s]_{p,q}p^{m-s+1}}{[m]_{p,q}^{2}} \\& \qquad {}+\frac {2[m+s]_{p,q}qp^{4m+2s-3}(p^{m+s}(1-x)+q^{m+s}x)}{[2]_{p,q}[m]_{p,q}^{2}(p(1-x)+qx)} \\& \qquad {}+\frac {p^{-2s}(p^{m+s}(1-x)+q^{m+s}x)(p^{m+s+1}(1-x)+q^{m+s+1}x)}{[3]_{p,q}[m]_{p,q}^{2}(p(1-x)+qx)} \biggr) \\& \quad = 0. \end{aligned}$$

Consequently, the proof is finished. □

Before mentioning local approximation properties, we will give two lemmas as follows.

Lemma 2

If f is a monotone increasing function, then the constructed operators \(K_{m,s}^{(p,q)}(f;x)\) are linear and positive.

Lemma 3

Let \(0< q< p\leq1\), \(0< u< v\), and \(\frac{1}{u }+\frac{1}{v }=1\). Then the operators \(K_{m,s}^{(p,q)} ( f;x )\) satisfy the following Hölder inequality:

$$ K_{m,s}^{(p,q)} \bigl( \vert fg \vert ;x \bigr) \leq \bigl( K_{m,s}^{(p,q)} \bigl( \vert f \vert ^{u };x \bigr) \bigr) ^{\frac{1}{u }} \bigl( K_{m,s}^{(p,q)} \bigl( \vert g \vert ^{v };x \bigr) \bigr) ^{\frac{1}{v }}. $$

Local approximation properties

Let f be a continuous function on \(C[0,s+1]\). The modulus of continuity of f is denoted by \(w(f,\sigma ) \) and given as

$$ w(f,\sigma)=\sup_{\overset{ \vert y-x \vert \leq \sigma}{x,y\in [ 0,1 ] }} \bigl\vert f ( y ) -f ( x ) \bigr\vert . $$
(20)

Then we know from the properties of modulus of continuity that for each \(\sigma>0\), we have

$$ \bigl\vert f ( y ) -f ( x ) \bigr\vert \leq w(f,\sigma) \biggl( \frac{ \vert y-x \vert }{\sigma}+1 \biggr), \quad x,y\in[0,1]. $$
(21)

And also, for \(f\in C[0,s+1]\) we have \(\lim_{\sigma\rightarrow 0^{+}}w(f,\sigma)=0\). First of all, we begin by giving the rate of convergence of the operators \(K_{m,s}^{(p,q)}(f;x)\) by using the modulus of continuity.

Theorem 2

Let the sequences \(p:= ( p_{m} ) \) and \(q:= ( q_{m} ) \), \(0< q_{m}< p_{m}\leq1\), satisfy the conditions \(p_{m}\rightarrow1\), \(q_{m}\rightarrow1\), \(p_{m}^{m}\rightarrow1\) and \(q_{m}^{m}\rightarrow1\) as \(m\rightarrow\infty\). Then for each \(f\in C [ 0,s+1 ]\),

$$ \bigl\lVert K_{m,s}^{(p,q)}f-f \bigr\lVert _{C[0,s+1]}\leq2\omega \bigl( f;\sigma_{m} ( x ) \bigr), $$

where

$$ \sigma_{m} ( x )=\sqrt{ K_{m,s}^{(p,q)} \bigl((t-x)^{2};x \bigr)} $$
(22)

and \(K_{m,s}^{(p,q)}((t-x)^{2};x)\) is as given by (19).

Proof

By the positivity and linearity of the operators \(K_{m,s}^{(p,q)}(f;x)\), we get

$$\begin{aligned} \bigl\vert K_{m,s}^{(p,q)}(f;x) -f ( x ) \bigr\vert =& \bigl\vert K_{m,s}^{(p,q)} \bigl(f(t)-f(x);x \bigr) \bigr\vert \\ \leq& K_{m,s}^{(p,q)} \bigl( \bigl\vert f ( t ) -f ( x ) \bigr\vert ;q;x \bigr). \end{aligned}$$

After that we apply (21) and obtain

$$\begin{aligned} \bigl\vert K_{m,s}^{(p,q)}(f;x) -f ( x ) \bigr\vert \leq& K_{m,s}^{(p,q)} \biggl( w(f,\sigma_{m}) \biggl( \frac{ \vert t-x \vert }{\sigma_{m} }+1 \biggr);x \biggr) \\ =& \frac{ w(f,\sigma_{m})}{\sigma_{m}}\sqrt{ K_{m,s}^{(p,q)} \bigl((t-x)^{2};x \bigr)}+w(f,\sigma_{m}) \\ =&w(f,\sigma_{m}) \biggl(1+\frac{1}{\sigma_{m}}\sqrt{ K_{m,s}^{(p,q)} \bigl((t-x)^{2};x \bigr)} \biggr). \end{aligned}$$
(23)

Then, taking supremum of the last equation, we have

$$\begin{aligned} \bigl\lVert K_{m,s}^{(p,q)}f-f \bigr\lVert =&\sup _{x \in[0,1] } \bigl\vert K_{m,s}^{(p,q)}(f;x) -f ( x ) \bigr\vert \\ \leq&w(f,\sigma_{m}) \biggl(1+\frac{1}{\sigma_{m}}\sqrt{ K_{m,s}^{(p,q)} \bigl((t-x)^{2};x \bigr)} \biggr). \end{aligned}$$

Choose

$$\begin{aligned} \sigma_{m} ( x ) =& \biggl\lbrace \biggl( \frac {q^{2}[m+l]_{p,q}[m+l-1]_{p,q}}{([m]_{p,q}+\beta)^{2}(p(1-x)+qx)}- \frac{2[m+l]_{p,q}}{[m]_{p,q}+\beta}+1 \biggr) x^{2} \\ &{}+ \biggl( -\frac{2\alpha}{[m]_{p,q}+\beta}+ \frac{[m+l]_{p,q} ( p^{m+l-1}+2\alpha)}{([m]_{p,q}+\beta)^{2}} \biggr) x + \biggl( \frac{\alpha}{[m]_{p,q}+\beta} \biggr)^{2} \biggr\rbrace ^{1/2}. \end{aligned}$$

Thus, we achieve

$$ \bigl\lVert K_{m,s}^{(p,q)}f-f \bigr\lVert _{C[0,s+1]}\leq2\omega \bigl( f;\sigma_{m} ( x ) \bigr). $$

This result completes the proof of the theorem. □

In what follows, by using Lipschitz functions, we will give the rate of convergence of the operators \(K_{m,s}^{(p,q)}(f;x)\). We remember that if the inequality

$$ \bigl\vert f ( y ) -f ( x ) \bigr\vert \leq M \vert y-x \vert ^{\alpha} ;\quad \forall x,y\in{}[ 0,1] $$
(24)

is satisfied, then f belongs to the class \(\mathrm{Lip}_{M} ( \alpha )\).

Theorem 3

Denote \(p:= ( p_{m} ) \) and \(q:= ( q_{m} ) \) satisfying \(0< q_{m}< p_{m}\leq1\). Then, for every \(f\in \mathrm{Lip}_{M} ( \alpha ) \), we have

$$ \bigl\lVert K_{m,s}^{(p,q)}f-f \bigr\rVert \leq M\sigma _{m}^{\alpha} ( x ), $$

where \(\sigma_{m}(x)\) is the same as in (22).

Proof

Let f belong to the class \(\mathrm{Lip}_{M} ( \alpha ) \) for some \(0<\alpha \leq1\). Using the monotonicity of the operators \(K_{m,s}^{(p,q)}(f;x) \) and (24), we obtain

$$\begin{aligned} \bigl\vert K_{m,s}^{(p,q)}(f;x) -f ( x ) \bigr\vert \leq& K_{m,s}^{(p,q)} \bigl( \bigl\vert f ( t ) -f ( x ) \bigr\vert ;x \bigr) \\ \leq&M K_{m,s}^{(p,q)} \bigl( \vert t-x \vert ^{\alpha};x \bigr). \end{aligned}$$

Taking \(p=\frac{2}{\alpha}\), \(q=\frac{2}{2-\alpha}\) and applying Hölder inequality yields

$$\begin{aligned} \bigl\vert K_{m,s}^{(p,q)} ( f;x ) -f ( x ) \bigr\vert \leq &M \bigl\{ K_{m,s}^{(p,q)} \bigl( (t-x ) ^{2};x \bigr) \bigr\} ^{\frac{\alpha}{2}} \\ \leq&M\sigma_{m}^{\alpha} ( x ). \end{aligned}$$

By choosing \(\sigma_{m} ( x )\) as in Theorem 2, we complete the proof as desired. □

Finally, in the light of Peetre-K functionals, we obtain the rate of convergence of the constructed operators \(K_{m,s}^{(p,q)}(f;x)\). We recall the properties of Peetre-K functionals, which are defined as

$$ K(f,\delta):=\inf_{g \in C^{2}[0,s+1] } \bigl\{ \Vert f-g \Vert _{C [ 0,s+1 ]}+\delta \Vert g \Vert _{C^{2} [ 0,s+1 ]} \bigr\} . $$

Here \(C^{2} [ 0,s+1 ]\) defines the space of the functions f such that \(f, f', f''\in C[0,s+1]\). The norm in this space is given by

$$ \Vert f \Vert _{C^{2} [ 0,s+1 ] }=\bigl\lVert f''\bigr\lVert _{C [ 0,s+1 ] } +\bigl\lVert f' \bigr\lVert _{C [ 0,s+1 ] }+\lVert f \lVert _{C [ 0,s+1 ] }. $$

Also we consider the second modulus of smoothness of \(f\in C[0,s+1]\), namely

$$ \omega_{2}(f,\delta):=\sup_{0< h< \delta} \sup _{x,x+h\in[0,s+1]} \bigl\vert f(x+2h)-2f(x+h)+f(x) \bigr\vert ,\quad \delta>0. $$

We know from [7] that for \(M>0\)

$$ K(f,\delta)\leq M\omega_{2}(f,\sqrt{\sigma}). $$

Before giving the main theorem, we present an auxiliary lemma, which will be used in the proof of the theorem.

Lemma 4

For any \(f\in C[0,s+1]\), we have

$$\begin{aligned} \bigl\vert K_{m,s}^{(p,q)}(f;x) \bigr\vert \leq \Vert f \Vert . \end{aligned}$$
(25)

Proof

$$\begin{aligned} \bigl\vert K_{m,s}^{(p,q)}(f;x) \bigr\vert =& \Biggl\vert \sum_{l=0}^{m+s}B_{m,l,s}^{p,q}(x) \int_{0}^{1}f \biggl( \frac{p[l]_{p,q}+q^{l}t}{p^{l-m}[m]_{p,q}} \biggr) \,d_{p,q}t \Biggr\vert \\ \leq& \sum_{l=0}^{m+s} B_{m,l,s}^{p,q}(x) \biggl\vert \int_{0}^{1}f \biggl( \frac{p[l]_{p,q}+q^{l}t}{p^{l-m}[m]_{p,q}} \biggr) \,d_{p,q}t \biggr\vert \\ \leq& \sum_{l=0}^{m+s} B_{m,l,s}^{p,q}(x) \int_{0}^{1} \biggl\vert f \biggl( \frac{p[l]_{p,q}+q^{l}t}{p^{l-m}[m]_{p,q}} \biggr) \biggr\vert \,d_{p,q}t \\ \leq& \Vert f \Vert K_{m,s}^{(p,q)} (1;x) \\ =& \Vert f \Vert . \end{aligned}$$

 □

Theorem 4

Let \(0< q_{m}< p_{m}\leq1\), \(m\in\mathbb{N}\) and \(f\in C[0,s+1] \). There exists a constant \(M>0\) such that

$$ \bigl\vert K_{m,s}^{(p,q)}(f;x)-f ( x ) \bigr\vert \leq M \omega_{2} \bigl(f,\alpha_{m}(x) \bigr)+\omega \bigl(f, \beta_{m}(x) \bigr), $$

where

$$\begin{aligned} \alpha_{m} ( x )= \sqrt{ K_{m,s}^{(p,q)} \bigl((t-x)^{2};x \bigr)+\frac{1}{2} \biggl( \frac {([2]_{p,q}[m+s]_{p,q}p^{1-s}-p^{m}+p^{-s}q^{m+s})x+p^{m} }{[2]_{p,q}[m]_{p,q}}-x \biggr)^{2} } \end{aligned}$$
(26)

and

$$\begin{aligned} \beta_{m}(x)= \frac{([2]_{p,q}[m+s]_{p,q}p^{1-s}-p^{m}+p^{-s}q^{m+s})x+p^{m} }{[2]_{p,q}[m]_{p,q}}-x. \end{aligned}$$
(27)

Proof

Define an auxiliary operator \(K_{m,s}^{*}\) as follows:

$$\begin{aligned} K_{m,s}^{*}(f;x)=K_{m,s}^{(p,q)}(f;x)-f \biggl( \frac{([2]_{p,q}[m+s]_{p,q}p^{1-s}-p^{m}+p^{-s}q^{m+s})x+p^{m} }{[2]_{p,q}[m]_{p,q}} \biggr)+f(x). \end{aligned}$$
(28)

From Lemma 1, we have

$$\begin{aligned}& K_{m,s}^{*}(1;x)=1, \\& \begin{aligned}[b] K_{m,s}^{*}(t-x;x)&= K_{m,s}^{(p,q)} \bigl((t-x);x \bigr)- \biggl( \frac {([2]_{p,q}[m+s]_{p,q}p^{1-s}-p^{m}+p^{-s}q^{m+s})x+p^{m} }{[2]_{p,q}[m]_{p,q}}-x \biggr)\hspace{-20pt} \\ &= \biggl( \frac{[m+s]_{p,q}}{p^{s-1}[m]_{p,q}} -\frac {p^{m}}{[2]_{p,q}[m]_{p,q}}+ \frac{q^{m+s}}{[2]_{p,q}[m]_{p,q}p^{s}}-1 \biggr)x+x \\ &\quad {}+\frac{p^{m}}{[2]_{p,q}[m]_{p,q}}-\frac {([2]_{p,q}[m+s]_{p,q}p^{1-s}-p^{m}+p^{-s}q^{m+s})x+p^{m} }{[2]_{p,q}[m]_{p,q}}\\ &=0. \end{aligned} \end{aligned}$$
(29)

Taylor’s expansion for a function \(g \in C^{2}[0,s+1]\) can be written as follows:

$$\begin{aligned} g(t)=g(x)+(t-x)g'(x)+ \int_{x}^{t}(t-u)g''(u)\,du, \quad t\in[0,1]. \end{aligned}$$
(30)

Then applying operator \(K_{m,s}^{*} \) to both sides of (30), we get

$$\begin{aligned} K_{m,s}^{*}(g;x) =&K_{m,s}^{*} \biggl(g(x)+(t-x)g'(x)+ \int_{x}^{t}(t-u)g''(u)\,du \biggr) \\ =&g(x)+K_{m,s}^{*} \bigl((t-x)g'(x);x \bigr)+K_{m,s}^{*} \biggl( \int_{x}^{t}(t-u)g''(u)\,du \biggr). \end{aligned}$$

So,

$$ K_{m,s}^{*}(g;x)-g(x)=g'(x)K_{m,s}^{*} \bigl((t-x);x \bigr)+K_{m,s}^{*} \biggl( \int_{x}^{t}(t-u)g''(u)\,du \biggr). $$

Using (29) and (28), we obtain

$$\begin{aligned}& K_{m,s}^{*}(g;x)-g(x) \\& \quad =K_{m,s}^{*} \biggl( \int_{x}^{t}(t-u)g''(u)\,du \biggr) \\& \quad = K_{m,s}^{(p,q)} \biggl( \int_{x}^{t}(t-u)g''(u)\,du \biggr) \\& \qquad {}- \int_{x}^{\frac{([2]_{p,q}[m+s]_{p,q}p^{1-s}-p^{m}+p^{-s}q^{m+s})x+p^{m} }{[2]_{p,q}[m]_{p,q}}} \biggl(\frac {([2]_{p,q}[m+s]_{p,q}p^{1-s}-p^{m}+p^{-s}q^{m+s})x+p^{m} }{[2]_{p,q}[m]_{p,q}} \\& \qquad {}-u \biggr)g''(u)\,du \\& \qquad {}+ \int_{x}^{x} \biggl(\frac {([2]_{p,q}[m+s]_{p,q}p^{1-s}-p^{m}+p^{-s}q^{m+s})x+p^{m} }{[2]_{p,q}[m]_{p,q}}-u \biggr)g''(u)\,du. \end{aligned}$$
(31)

Moreover,

$$ \biggl\vert \int_{x}^{t}(t-u)g''(u)\,du \biggr\vert \leq \int_{x}^{t} \vert t-u \vert \bigl\vert g''(u) \bigr\vert \,du \leq \bigl\Vert g'' \bigr\Vert \int_{x}^{t} \vert t-u \vert \,du \leq (t-x)^{2} \bigl\Vert g'' \bigr\Vert $$
(32)

and

$$\begin{aligned}& \biggl\vert \int_{x}^{\frac{([2]_{p,q}[m+s]_{p,q}p^{1-s}-p^{m}+p^{-s}q^{m+s})x+p^{m} }{[2]_{p,q}[m]_{p,q}}} \biggl(\frac {([2]_{p,q}[m+s]_{p,q}p^{1-s}-p^{m}+p^{-s}q^{m+s})x+p^{m} }{[2]_{p,q}[m]_{p,q}} \\& \qquad {}-u \biggr)g''(u)\,du \biggr\vert \\& \quad \leq \bigl\Vert g'' \bigr\Vert \int_{x}^{\frac{([2]_{p,q}[m+s]_{p,q}p^{1-s}-p^{m}+p^{-s}q^{m+s})x+p^{m} }{[2]_{p,q}[m]_{p,q}}} \biggl(\frac {([2]_{p,q}[m+s]_{p,q}p^{1-s}-p^{m}+p^{-s}q^{m+s})x+p^{m} }{[2]_{p,q}[m]_{p,q}} \\& \qquad {}-u \biggr)\,du \\& \quad = \frac{ \Vert g'' \Vert }{2} \biggl( \frac {([2]_{p,q}[m+s]_{p,q}p^{1-s}-p^{m}+p^{-s}q^{m+s})x+p^{m} }{[2]_{p,q}[m]_{p,q}}-x \biggr)^{2}. \end{aligned}$$
(33)

Let us employ (32) and (33) when taking the absolute value of (31). We obtain

$$\begin{aligned} \bigl\vert K_{m,s}^{*}(g;x)-g(x) \bigr\vert \leq& \bigl\Vert g'' \bigr\Vert K_{m,s}^{(p,q)} \bigl((t-x)^{2};x \bigr) \\ &{}+\frac{ \Vert g'' \Vert }{2} \biggl( \frac {([2]_{p,q}[m+s]_{p,q}p^{1-s}-p^{m}+p^{-s}q^{m+s})x+p^{m} }{[2]_{p,q}[m]_{p,q}}-x \biggr)^{2} \\ =& \bigl\Vert g'' \bigr\Vert \alpha_{m}^{2}(x), \end{aligned}$$

where

$$\begin{aligned}& \alpha_{m} ( x ) \\& \quad =\sqrt{ K_{m,s}^{(p,q)} \bigl((t-x)^{2};x \bigr)+\frac{1}{2} \biggl( \frac {([2]_{p,q}[m+s]_{p,q}p^{1-s}-p^{m}+p^{-s}q^{m+s})x+p^{m} }{[2]_{p,q}[m]_{p,q}}-x \biggr)^{2} }. \end{aligned}$$
(34)

We now give an upper bound for the auxiliary operator \(K_{m,l,p,q}^{*}(f;x)\). From Lemma 4 we get

$$\begin{aligned} \bigl\vert K_{m,s}^{*}(f;x) \bigr\vert =& \biggl\vert K_{m,s}^{(p,q)}(f;x)-f \biggl( \frac {([2]_{p,q}[m+s]_{p,q}p^{1-s}-p^{m}+p^{-s}q^{m+s})x+p^{m} }{[2]_{p,q}[m]_{p,q}} \biggr)+f(x) \biggr\vert \\ \leq& \bigl\vert K_{m,s}^{(p,q)}(f;x) \bigr\vert + \biggl\vert f \biggl( \frac {([2]_{p,q}[m+s]_{p,q}p^{1-s}-p^{m}+p^{-s}q^{m+s})x+p^{m} }{[2]_{p,q}[m]_{p,q}} \biggr) \biggr\vert + \bigl\vert f(x) \bigr\vert \\ \leq&3 \Vert f \Vert . \end{aligned}$$

Accordingly,

$$\begin{aligned}& \begin{aligned} &\bigl\vert K_{m,s}^{(p,q)}(f;x)-f(x) \bigr\vert \\ &\quad = \biggl\vert K_{m,s}^{*}(f;x)-f(x)+f \biggl( \frac {([2]_{p,q}[m+s]_{p,q}p^{1-s}-p^{m}+p^{-s}q^{m+s})x+p^{m} }{[2]_{p,q}[m]_{p,q}} \biggr)-f(x) \\ &\qquad {} \mp g(x)\mp K_{m,s}^{*}(g;x) \biggr\vert , \end{aligned} \\& \begin{aligned}[b] &\bigl\vert K_{m,s}^{(p,q)}(f;x)-f(x) \bigr\vert \\ &\quad \leq \bigl\vert K_{m,s}^{*}(f-g;x)-(f-g) (x) \bigr\vert + \bigl\vert K_{m,s}^{*}(g;x)-g(x) \bigr\vert \\ &\qquad {}+ \biggl\vert f \biggl( \frac {([2]_{p,q}[m+s]_{p,q}p^{1-s}-p^{m}+p^{-s}q^{m+s})x+p^{m} }{[2]_{p,q}[m]_{p,q}} \biggr) -f(x) \biggr\vert \\ &\quad \leq4 \Vert f-g \Vert + \bigl\Vert g'' \bigr\Vert \alpha_{m}^{2}(x)+\omega \bigl(f,\beta_{m}(x) \bigr) \biggl(\frac{ ( \frac {([2]_{p,q}[m+s]_{p,q}p^{1-s}-p^{m}+p^{-s}q^{m+s})x+p^{m} }{[2]_{p,q}[m]_{p,q}}-x )}{\beta_{m}(x)}+1 \biggr)\hspace{-20pt} \\ &\quad =4 \Vert f-g \Vert + \bigl\Vert g'' \bigr\Vert \alpha_{m}^{2}(x) \\ &\qquad {}+2\omega \biggl(f, \biggl( \frac {([2]_{p,q}[m+s]_{p,q}p^{1-s}-p^{m}+p^{-s}q^{m+s})x+p^{m} }{[2]_{p,q}[m]_{p,q}}-x \biggr) \biggr), \end{aligned} \end{aligned}$$
(35)

where

$$ \beta_{m}(x)= \frac{([2]_{p,q}[m+s]_{p,q}p^{1-s}-p^{m}+p^{-s}q^{m+s})x+p^{m} }{[2]_{p,q}[m]_{p,q}}-x. $$
(36)

Finally, for all \(g\in C^{2}[0,s+1]\), taking the infimum of (35), we get

$$ \bigl\vert K_{m,s}^{(p,q)}(f;x)-f(x) \bigr\vert \leq4K \bigl(f,\alpha_{m}^{2}(x) \bigr)+\omega \bigl(f, \beta_{m}(x) \bigr). $$
(37)

Consequently, using the property of Peetre-K functional, we obtain

$$ \bigl\vert K_{m,s}^{(p,q)}(f;x)-f(x) \bigr\vert \leq M \omega_{2} \bigl(f,\alpha_{m}(x) \bigr)+\omega \bigl(f, \beta_{m}(x) \bigr). $$
(38)

This completes the proof. □

Graphical illustrations

In this section, we illustrate an approximation of the operators \(K_{m,s}^{(p,q)}\) for a function \(f(x)\) by employing Matlab codes. Let us specially choose

$$f(x)= \frac{1}{96}\tan\biggl(\frac{x}{16}\biggr) \biggl( \frac{x}{8}\biggr)^{2}\biggl(1-\frac{x}{4} \biggr)^{3}, $$

and take \(p=0.8\), \(q=0.7\) and \(s=5\).

Algorithm 1

Algorithm 2

Initially, we discuss the error estimates of the Kantorovich type Lupaş–Schurer operators based on \((p, q)\)-integers for different values of x and m in Table 1 by using Algorithm 1.

Table 1 Error estimates for different values of x when \(s=5\), \(p=0.8\) and \(q=0.7\)

And then, we illustrate the convergence of the \((p, q)\)-Lupaş–Schurer–Kantorovich operators \(K_{m,s}^{(p,q)}(f;x)\) for the selected function \(f(x)= \frac{1}{96}\tan(\frac{x}{16})(\frac {x}{8})^{2}(1-\frac{x}{4})^{3}\) in Fig. 1 for several values of m by using Algorithm 2. Furthermore, we give the error estimates in Table 2 in order to indicate that the \((p, q)\)-analogue Lupaş–Schurer operators [14] converge and then plot Fig. 2. It can be clearly seen that the \((p, q)\)-Lupaş–Schurer–Kantorovich operators converge faster than the \((p, q)\)-analogue Lupaş–Schurer operators.

Figure 1
figure 1

Convergence of \((p, q)\)-analogue Lupaş–Schurer–Kantorovich operators \(K_{m,s}^{(p,q)}(f;x)\) for various values of \(p,1\) and m with fixed \(s=5\)

Figure 2
figure 2

Convergence of the \((p, q)\)-analogue Lupaş–Schurer operators \(L_{m,l}^{p,q}(f;x)\) with fixed \(l=5\) for various values of p and q

Table 2 Error estimates of \((p, q)\)-Lupaş–Schurer operators for various values of x

Conclusion

In this paper, we constructed a new kind of Lupaş operators based on \((p, q)\)-integers to provide a better error estimation. Firstly, we investigated some local approximation results by the help of the well-known Korovkin theorem. Also, we calculated the rate of convergence of the constructed operators employing the modulus of continuity, by using Lipschitz functions and then with the help of Peetre’s K-functional. Additionally, we presented a table of error estimates of the \((p, q)\)-Lupaş–Schurer–Kantorovich operators for a certain function. Finally, we compared the convergence of the new operator to that of the \((p, q)\)-analogue of Lupaş–Schurer operator.

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Kanat, K., Sofyalıoğlu, M. Approximation by \((p,q)\)-Lupaş–Schurer–Kantorovich operators. J Inequal Appl 2018, 263 (2018). https://doi.org/10.1186/s13660-018-1858-9

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Keywords

  • Lupaş operators
  • \((p,q)\)-integers
  • Rate of convergence
  • Local approximation
  • Korovkin’s approximation theorem