Open Access

On \((p,q)\)-Szász-Mirakyan operators and their approximation properties

Journal of Inequalities and Applications20172017:196

https://doi.org/10.1186/s13660-017-1467-z

Received: 2 May 2017

Accepted: 3 August 2017

Published: 24 August 2017

Abstract

In the present paper, we introduce a new modification of Szász-Mirakyan operators based on \((p, q)\)-integers and investigate their approximation properties. We obtain weighted approximation and Voronovskaya-type theorem for new operators.

Keywords

q-integers \((p, q)\)-integers q-Szász-Mirakyan operators \((p, q)\)-Szász-Mirakyan operators weighted approximation Voronovskaya theorem

MSC

41A10 41A25 41A36

1 Introduction and preliminaries

In the last two decades, there has been intensive research on the approximation of functions by positive linear operators introduced by using q-calculus. Lupas [1] was the first who used q-calculus to define q-Bernstein polynomials, and later Phillips [2] proposed a generalization of Bernstein polynomials based on q-integers. Very recently, Mursaleen et al. applied \((p,q)\)-calculus in approximation theory and introduced the first \((p,q)\)-analogue of Bernstein operators [3]. They investigated the uniform convergence and convergence rate of the operators and also obtained a Voronovskaya-type theorem. Also, \((p,q)\)-analogues of Bernstein-Stancu operators [4], Bleimann-Butzer-Hahn operators [5], and Bernstein-Schurer operarors [6] were defined and their approximation properties were investigated. Most recently, the \((p,q)\)-analogues of some more operators were defined and their approximation properties were studied in [717], and [18]. In this paper, we introduce a \((p,q)\)-analogue of Szász-Mirakyan operators. Let us recall some notation and definitions of \((p,q)\)-calculus. Let \(0< q< p\leq1\). For nonnegative integers k and n such that \(n\geq k\geq0\), the \((p,q)\)-integer, \((p,q)\)-factorial, and \((p,q)\)-binomial are respectively defined by
$$ \begin{gathered} {}[ k]_{p,q}:=\frac{p^{k}-q^{k}}{p-q}, \\ {}[ k]_{p,q}!:=\left \{ \textstyle\begin{array}{l@{\quad}l} {}[ k]_{p,q}[k-1]_{p,q}\cdots1,& k\geq1, \\ 1, & k=0 ,\end{array}\displaystyle \right . \end{gathered} $$
and
$$ \left [ \textstyle\begin{array}{c} n \\ k\end{array}\displaystyle \right ] _{p,q}:= \frac{[n]_{p,q}!}{[k]_{p,q}![n-k]_{p,q}!}. $$
In the case of \(p=1\), these notations reduce to q-analogues, and we can easily see that \([n]_{p,q}=p^{n-1}[n]_{q/p}\). Further, the \((p,q)\)-power basis is defined by
$$ (x\oplus a)_{p,q}^{n}:=(x+a) (px+qa) \bigl(p^{2}x+q^{2}a \bigr)\cdots \bigl(p^{n-1}x+q^{n-1}a \bigr) $$
and
$$ (x\ominus a)_{p,q}^{n}:=(x-a) (px-qa) \bigl(p^{2}x-q^{2}a \bigr)\cdots \bigl(p^{n-1}x-q^{n-1}a \bigr). $$
Also the \((p,q)\)-derivative of a function f, denoted by \(D_{p,q}f\), is defined by
$$ (D_{p,q}f) (x):=\frac{f(px)-f(qx)}{(p-q)x},\quad x\neq 0,\qquad(D_{p,q}f) (0):=f^{{\prime}}(0) $$
provided that f is differentiable at 0. The formula for the \((p,q)\)-derivative of a product is
$$ D_{p,q} \bigl(u(x)v(x) \bigr):=D_{p,q} \bigl(u(x) \bigr)v(qx)+D_{p,q} \bigl(v(x) \bigr)u(qx). $$
For more details on \((p,q)\)-calculus, we refer the readers to [19, 20] and the references therein. There are two \((p,q)\)-analogues of the exponential function:
$$ e_{p,q}(x)=\sum_{n=0}^{\infty} \frac{p^{\frac{n(n-1)}{2}}x^{n}}{[n]_{p,q}!} $$
(1.1)
and
$$ E_{p,q}(x)=\sum_{n=0}^{\infty} \frac{q^{\frac{n(n-1)}{2}}x^{n}}{[n]_{p,q}!} $$
which satisfy the equality \(e_{p,q}(x)E_{p,q}(-x)=1\). For \(p=1\), \(e_{p,q}(x)\) and \(E_{p,q}(x)\) reduce to the q-exponential functions. Here we note that the interval of convergence of \(e_{p,q}(x)\) is \(| x|<1/(p-q)\) for \(| p|<1\) and \(| q|<1\), and series (1.1) converges for all \(x\in\mathbb{R}\), \(| p|<1\), and \(| q|<1\).

2 Construction of operators and auxiliary results

We first define the analogue of Szász-Mirakyan operators via \((p, q)\)-calculus as follows.

Definition 2.1

Let \(0< q< p\leq1\) and \(n\in\mathbb{N}\). For \(f:[0,\infty)\rightarrow\mathbb{R}\), we define the \((p, q)\)-analogue of Szász-Mirakyan operators by
$$ {S}_{n,p,q}(f;x)= \sum_{k=0}^{\infty} \frac{p^{ \frac{k(k-1)}{2}}}{q^{ \frac{k(k-1)}{2}}}\frac{([n]_{p,q}x)^{k} }{ [k]_{p,q}! }e_{p,q} \bigl(-[n]_{p,q}q ^{-k}x \bigr) f \biggl(\frac {[k]_{p,q}}{p^{k-1}[n]_{p,q}} \biggr). $$
(2.1)

Operators (2.1) are linear and positive. For \(p=1\), they turn out to be the q-Szász-Mirakyan operators defined in [21].

Lemma 2.1

Let \(0< q< p\leq1\) and \(n\in\mathbb{N}\). We have
$$ {S}_{n,p,q} \bigl(t^{m+1};x \bigr)= \sum _{j=0}^{m} \left ( \textstyle\begin{array}{c} m \\ j\end{array}\displaystyle \right ) \frac{q^{j}x }{p^{j} [n]_{p,q}^{m-j} }{S}_{n,p,q} \bigl(t^{j};q^{-1}x \bigr). $$
(2.2)

Proof

Using the identity
$$ [k+1]_{p,q}=p^{k}+q[k]_{p,q}, $$
we can write
$$ \begin{aligned} {S}_{n,p,q} \bigl(t^{m+1};x \bigr) ={}&\sum _{k=0}^{\infty} \frac{p^{ \frac{k(k-1)}{2}}}{q^{ \frac{k(k-1)}{2}}} \frac {([n]_{p,q}x)^{k} }{ [k]_{p,q}! } \biggl(\frac{[k]_{p,q}}{p^{k-1}[n]_{p,q}} \biggr)^{m+1} e_{p,q} \bigl(-[n]_{p,q}q ^{-k}x \bigr) \\ ={}& \sum_{k=0}^{\infty} \frac{p^{k} p^{ \frac{k(k-1)}{2}}}{q^{k} q^{ \frac{k(k-1)}{2}}} \frac{([n]_{p,q}x)^{k} }{ [k]_{p,q}! } \frac{[k+1]_{p,q}^{m} x}{p^{k(m+1)}[n]_{p,q}^{m}}e_{p,q} \bigl(-[n]_{p,q}q ^{-(k+1)}x \bigr) \\ ={}& \sum_{k=0}^{\infty} \frac{p^{k} p^{ \frac{k(k-1)}{2}}}{q^{k} q^{ \frac{k(k-1)}{2}}} \frac{([n]_{p,q}x)^{k} }{ [k]_{p,q}! } \frac{[k+1]_{p,q}^{m} x}{p^{km+k}[n]_{p,q}^{m}}e_{p,q} \bigl(-[n]_{p,q}q ^{-(k+1)}x \bigr) \\ ={}& \sum_{k=0}^{\infty} \frac{ p^{ \frac{k(k-1)}{2}}}{q^{k} q^{ \frac{k(k-1)}{2}}} \frac{([n]_{p,q}x)^{k} }{ [k]_{p,q}! } \frac{(p^{k}+q[k]_{p,q})^{m} x}{p^{km}[n]_{p,q}^{m}}e_{p,q} \bigl(-[n]_{p,q}q ^{-(k+1)}x \bigr) \\ ={}& \sum_{k=0}^{\infty} \frac{ x }{ p^{km} [n]_{p,q}^{m} } \frac{ p^{ \frac{k(k-1)}{2}}}{q^{k} q^{ \frac {k(k-1)}{2}}}\frac{([n]_{p,q}x)^{k} }{ [k]_{p,q}! }\\ &\times\sum_{j=0}^{m} \left ( \textstyle\begin{array}{c} m \\ j\end{array}\displaystyle \right )p^{k(m-j)}q^{j}[k]_{p,q}^{j}e_{p,q} \bigl(-[n]_{p,q}q ^{-(k+1)}x \bigr) \\ ={}& \sum_{j=0}^{m}\left ( \textstyle\begin{array}{c} m \\ j\end{array}\displaystyle \right ) \frac{ q^{j}x }{ p^{j} [n]_{p,q}^{m-j} }\\ &\times \sum _{k=0}^{\infty} \frac{ [k]_{p,q}^{j} }{ p^{j(k-1)} [n]_{p,q}^{j} } \frac{ p^{ \frac {k(k-1)}{2}}}{q^{k} q^{ \frac{k(k-1)}{2}}} \frac{([n]_{p,q}x)^{k} }{ [k]_{p,q}! } e_{p,q} \bigl(-[n]_{p,q}q ^{-(k+1)}x \bigr) \\ ={}& \sum_{j=0}^{m} \left ( \textstyle\begin{array}{c} m \\ j\end{array}\displaystyle \right ) \frac{q^{j}x }{p^{j} [n]_{p,q}^{m-j} } {S}_{n,p,q} \bigl(t^{j};q^{-1}x \bigr), \end{aligned} $$
as desired. □

Lemma 2.2

Let \(0< q< p\leq1\) and \(n\in\mathbb{N}\). We have
  1. (i)

    \({S}_{n,p,q}(1;x)=1\),

     
  2. (ii)

    \({S}_{n,p,q}(t;x)=x\),

     
  3. (iii)

    \({S}_{n,p,q}(t^{2};x)= \frac{x^{2}}{p}+\frac{ x}{[n]_{p,q} } \),

     
  4. (iv)

    \({S}_{n,p,q}(t^{3};x)= \frac{x^{3}}{p^{3}}+ \frac{ 2p+q}{p^{2}[n]_{p,q} }x^{2} + \frac{ x}{[n]_{p,q}^{2} }\),

     
  5. (v)

    \({S}_{n,p,q}(t^{4};x)= \frac{x^{4}}{p^{6}}+ \frac{3p^{2}+ 2pq+q^{2}}{p^{5}[n]_{p,q} }x^{3} + \frac{3p^{2}+ 3pq+q^{2}}{p^{3}[n]_{p,q}^{2} }x^{2}+ \frac{ x}{[n]_{p,q}^{3} }\).

     

Proof

Since the proof of each equality uses the same method, we give the proof for only last three equalities. Using (2.2), we get
  1. (iii)
    $$\begin{aligned} {S}_{n,p,q} \bigl(t^{2};x \bigr) ={}& \sum _{k=0}^{\infty} \frac{ p^{ \frac{k(k-1)}{2}}}{ q^{ \frac{k(k-1)}{2}}}\frac{([n]_{p,q}x)^{k}}{ [k]_{p,q}! } \frac {[k]_{p,q}^{2}}{p^{2k-2}[n]_{p,q}^{2}}e_{p,q} \bigl(-[n]_{p,q}q ^{-k}x \bigr) \\ ={}& \sum_{k=0}^{\infty} \frac{ p^{k}p^{ \frac{k(k-1)}{2}}}{ q^{k}q^{ \frac{k(k-1)}{2}}} \frac{([n]_{p,q}x)^{k}}{ [k]_{p,q}! } \frac{[k+1]_{p,q}x}{p^{2k}[n]_{p,q}}e_{p,q} \bigl(-[n]_{p,q}q ^{-(k+1)}x \bigr) \\ ={}& \sum_{k=0}^{\infty}\frac{ p^{k}p^{ \frac{k(k-1)}{2}}}{ q^{k}q^{ \frac{k(k-1)}{2}}} \frac{([n]_{p,q}x)^{k}}{ [k]_{p,q}! }\frac{p^{k}x}{p^{2k}[n]_{p,q}}e_{p,q} \bigl(-[n]_{p,q}q ^{-(k+1)}x \bigr) \\ &+ \sum_{k=0}^{\infty}\frac{ p^{ \frac{k(k-1)}{2}}}{ q^{k}q^{ \frac{k(k-1)}{2}}} \frac{([n]_{p,q}x)^{k}}{ [k]_{p,q}! }\frac{q [k]_{p,q}x }{p^{k}[n]_{p,q}}e_{p,q} \bigl(-[n]_{p,q}q ^{-(k+1)}x \bigr) \\ ={}& \frac{ x}{[n]_{p,q} }+ \sum_{k=0}^{\infty} \frac{ p^{ \frac{k(k-1)}{2}}}{ q^{2k}q^{ \frac {k(k-1)}{2}}}\frac{([n]_{p,q}x)^{k}}{ [k]_{p,q}! }\frac{x^{2} }{p} e_{p,q} \bigl(-[n]_{p,q}q ^{-(k+2)}x \bigr) \\ ={}& \frac{x^{2}}{p}+\frac{ x}{[n]_{p,q} }. \end{aligned}$$
     
  2. (iv)
    $$\begin{aligned} {S}_{n,p,q} \bigl(t^{3};x \bigr) ={}& \sum _{k=0}^{\infty} \frac{ p^{ \frac{k(k-1)}{2}}}{ q^{ \frac{k(k-1)}{2}}}\frac{([n]_{p,q}x)^{k}}{ [k]_{p,q}! } \frac {[k]_{p,q}^{3}}{p^{3k-3}[n]_{p,q}^{3}}e_{p,q} \bigl(-[n]_{p,q}q ^{-k}x \bigr) \\ ={}& \sum_{k=0}^{\infty} \frac{ p^{ \frac{k(k-1)}{2}}}{ q^{k}q^{ \frac{k(k-1)}{2}}} \frac{([n]_{p,q}x)^{k}}{ [k]_{p,q}! } \frac {(p^{2k}+2p^{k}q[k]_{p,q}+q^{2}[k]_{p,q}^{2})}{p^{2k}[n]_{p,q}^{2}}\\ &\times x e_{p,q} \bigl(-[n]_{p,q}q ^{-(k+1)}x \bigr) \\ ={}& \sum_{k=0}^{\infty} \frac{ p^{ \frac{k(k-1)}{2}}}{q^{k} q^{ \frac{k(k-1)}{2}}} \frac{([n]_{p,q}x)^{k}}{ [k]_{p,q}! } \frac{x}{[n]_{p,q}^{2}}e_{p,q} \bigl(-[n]_{p,q}q ^{-(k+1)}x \bigr) \\ & +\sum_{k=0}^{\infty} \frac{ p^{ \frac{k(k-1)}{2}}}{ q^{k}q^{ \frac{k(k-1)}{2}}} \frac{([n]_{p,q}x)^{k}}{ [k]_{p,q}! } \frac{2q[k]_{p,q}}{p^{k}[n]_{p,q}^{2}}x e_{p,q} \bigl(-[n]_{p,q}q ^{-(k+1)}x \bigr) \\ &+ \sum_{k=0}^{\infty} \frac{ p^{ \frac{k(k-1)}{2}}}{ q^{k}q^{ \frac{k(k-1)}{2}}} \frac{([n]_{p,q}x)^{k}}{ [k]_{p,q}! } \frac{q^{2}[k]_{p,q}^{2}}{p^{2k}[n]_{p,q}^{2}}x e_{p,q} \bigl(-[n]_{p,q}q ^{-(k+1)}x \bigr) \\ ={}& \frac{ x}{[n]_{p,q}^{2} } +\frac{ 2x^{2}}{p[n]_{p,q} }\\ &+ \sum_{k=0}^{\infty} \frac{p^{k} p^{ \frac{k(k-1)}{2}}}{ q^{2k}q^{ \frac{k(k-1)}{2}}}\frac{([n]_{p,q}x)^{k}}{ [k]_{p,q}! } \frac{q x^{2} (p^{k}+ q[k]_{p,q})}{p^{2k+2}[n]_{p,q}} e_{p,q} \bigl(-[n]_{p,q}q ^{-(k+2)}x \bigr) \\ ={}& \frac{ x}{[n]_{p,q}^{2} } +\frac{ 2x^{2}}{p[n]_{p,q} }+ \frac{ qx^{2}}{p^{2}[n]_{p,q} } \\ &+ \sum _{k=0}^{\infty} \frac{ p^{ \frac{k(k-1)}{2}}}{ q^{2k}q^{ \frac{k(k-1)}{2}}}\frac{([n]_{p,q}x)^{k}}{ [k]_{p,q}! } \frac{q^{2} x^{2} [k]_{p,q}}{p^{k+2}[n]_{p,q}} e_{p,q} \bigl(-[n]_{p,q}q ^{-(k+2)}x \bigr) \\ ={}& \frac{x^{3}}{p^{3}}+ \frac{ 2p+q}{p^{2}[n]_{p,q} }x^{2} + \frac{ x}{[n]_{p,q}^{2} }. \end{aligned}$$
     
  3. (v)
    $$\begin{aligned} {S}_{n,p,q} \bigl(t^{4};x \bigr) ={}& \sum _{k=0}^{\infty} \frac{ p^{ \frac{k(k-1)}{2}}}{ q^{ \frac{k(k-1)}{2}}}\frac{([n]_{p,q}x)^{k}}{ [k]_{p,q}! } \frac {[k]_{p,q}^{4}}{p^{4k-4}[n]_{p,q}^{4}}e_{p,q} \bigl(-[n]_{p,q}q ^{-k}x \bigr) \\ ={}& \sum_{k=0}^{\infty} \frac{ p^{ \frac{k(k-1)}{2}}}{ q^{k}q^{ \frac{k(k-1)}{2}}} \frac{([n]_{p,q}x)^{k}}{ [k]_{p,q}! } \frac{(p^{3k}+3p^{2k}q[k]_{p,q}+3p^{k}q^{2}[k]_{p,q}^{2}+q^{3}[k]_{p,q}^{3})}{ p^{3k}[n]_{p,q}^{3}}\\ &\times x e_{p,q} \bigl(-[n]_{p,q}q ^{-(k+1)}x \bigr) \\ ={}& \frac{ x}{[n]_{p,q}^{3} } +\frac{ 3x^{2}}{p[n]_{p,q}^{2} }+ \frac{ 3qx^{2}}{p^{2}[n]_{p,q}^{2} }+ \frac{ 3x^{3}}{p^{3}[n]_{p,q} } \\ &+ \sum_{k=0}^{\infty} \frac{ p^{ \frac{k(k-1)}{2}}}{ q^{2k}q^{ \frac{k(k-1)}{2}}} \frac{([n]_{p,q}x)^{k}}{ [k]_{p,q}! } \frac{q^{2} x^{2} (p^{2k}+ 2p^{k}q[k]_{p,q}+q^{2} [k]_{p,q}^{2} )}{p^{2k+3}[n]_{p,q}^{2}}\\ &\times e_{p,q} \bigl(-[n]_{p,q}q ^{-(k+2)}x \bigr) \\ ={}& \frac{ x}{[n]_{p,q}^{3} } +\frac{ 3x^{2}}{p[n]_{p,q}^{2} }+ \frac{ 3qx^{2}}{p^{2}[n]_{p,q}^{2} }+ \frac{ 3x^{3}}{p^{3}[n]_{p,q} }+\frac{ q^{2}x^{2}}{p^{3}[n]_{p,q}^{2} } + \frac{ 2qx^{3}}{p^{4}[n]_{p,q} } \\ &+ \sum_{k=0}^{\infty} \frac{ p^{ \frac{k(k-1)}{2}}}{ q^{3k}q^{ \frac{k(k-1)}{2}}} \frac{([n]_{p,q}x)^{k}}{ [k]_{p,q}! } \frac{q^{2} x^{3} (p^{k}+ q[k]_{p,q} )}{p^{k+5}[n]_{p,q}} e_{p,q} \bigl(-[n]_{p,q}q ^{-(k+3)}x \bigr) \\ ={}& \frac{x^{4}}{p^{6}}+ \frac{3p^{2}+ 2pq+q^{2}}{p^{5}[n]_{p,q} }x^{3} + \frac{3p^{2}+ 3pq+q^{2}}{p^{3}[n]_{p,q}^{2} }x^{2}+ \frac{ x}{[n]_{p,q}^{3} }. \end{aligned}$$
     
 □

Corollary 2.1

Using Lemma 2.2, we immediately have the following explicit formulas for the central moments:
$$\begin{aligned}& {S}_{n,p,q} \bigl((t-x)^{2};x \bigr) = \frac{ x}{[n]_{p,q} }+ \biggl(\frac{1}{p}-1 \biggr)x^{2}, \end{aligned}$$
(2.3)
$$\begin{aligned}& {S}_{n,p,q} \bigl((t-x)^{3};x \bigr)= \frac{ x}{[n]_{p,q}^{2} }+ \frac {2p+q-3p^{2}}{p^{2}[n]_{p,q}} x^{2} +\frac{1-3p^{2}+2p^{3}}{p^{3}} x^{3}, \end{aligned}$$
(2.4)
$$\begin{aligned}& \begin{aligned}[b] {S}_{n,p,q} \bigl((t-x)^{4};x \bigr) ={}& \frac{ x}{[n]_{p,q}^{3} }+ \frac{3p^{2}+3pq+ q^{2}-4p^{3}}{p^{3}[n]_{p,q}^{2}} x^{2} \\ &+\frac{3p^{2}+2pq+ q^{2}-8p^{4}-4p^{3}q+6p^{5}}{p^{5}[n]_{p,q}} x^{3}\\ &+ \frac{1-4p^{3}+6p^{5}-3p^{6}}{p^{6}}x^{4}. \end{aligned} \end{aligned}$$
(2.5)

Remark 2.1

For \(q\in(0, 1)\) and \(p\in(q, 1]\) we easily see that \(\lim_{n\rightarrow\infty}[n]_{p,q}=\frac{1}{p-q}\). Hence, operators (2.1) are not approximation process with above form. To study convergence properties of the sequence of \((p, q)\)-Szász operators, we assume that \(q = (q_{n})\) and \(p = (p_{n})\) are such that \(0 < q_{n} < p_{n} \leq1\) and \(q_{n} \rightarrow1\), \(p_{n} \rightarrow1\), \(q_{n} ^{n} \rightarrow a\), \(p^{n}_{n} \rightarrow b\) as \(n \rightarrow\infty\). We also assume that
$$ \begin{gathered} \lim_{n\rightarrow\infty}[n]_{p_{n},q_{n}} \biggl( \frac{1}{p_{n}}-1 \biggr) = \alpha, \\ \lim_{n\rightarrow\infty}[n]_{p_{n},q_{n}} \frac {1-3p^{2}_{n}+2p^{3}_{n}}{p^{3}_{n}} = \gamma, \\ \lim_{n\rightarrow\infty}[n]_{p_{n},q_{n}} \frac{1-4p^{3}_{n}+6p^{5}_{n}-3p^{6}_{n}}{p^{6}_{n}} =\beta. \end{gathered} $$

It is natural to ask whether such sequences \((q_{n})\) and \((p_{n})\) exist. For example, let \(c, d \in\mathbb{R^{+}}\) be such that \(c > d\). If we choose \(q_{n}=\frac{n}{n+c}\) and \(p_{n}=\frac{n}{n+d}\), then \(q_{n} \rightarrow1\), \(p_{n} \rightarrow1\), \(q^{n}_{n} \rightarrow e^{-c}\), \(p^{n}_{n} \rightarrow e^{-d}\), and \(\lim_{n\rightarrow\infty}[n]_{p,q}=\infty\) as \(n \rightarrow \infty\). Also, we have \(\alpha=\frac{a(e^{-d}- e^{-c}) }{d-c}\), \(\gamma=e^{-d}- e^{-c}\), \(\beta=0\).

Corollary 2.2

According to Remark 2.1, we immediately have
$$\begin{aligned}& \lim_{n\rightarrow\infty}[n]_{p_{n},q_{n}}{S}_{n,p_{n},q_{n}} \bigl((t-x)^{2};x \bigr) = x+\alpha x^{2}, \end{aligned}$$
(2.6)
$$\begin{aligned}& \lim_{n\rightarrow\infty}[n]_{p_{n},q_{n}}{S}_{n,p_{n},q_{n}} \bigl((t-x)^{3};x \bigr)= \gamma x^{3}, \end{aligned}$$
(2.7)
$$\begin{aligned}& \lim_{n\rightarrow\infty}[n]_{p_{n},q_{n}}{S}_{n,p_{n},q_{n}} \bigl((t-x)^{4};x \bigr) = \beta x^{4}. \end{aligned}$$
(2.8)

3 Direct results

In this section, we present a local approximation theorem for the operators \(S_{n,p,q}\). By \(C_{B}[0,\infty)\) we denote the space of real-valued continuous and bounded functions f defined on the interval \([0,\infty)\). The norm \(\|\cdot\|\) on the space \(C_{B}[0,\infty)\) is given by
$$ \| f\|=\sup_{0\leq x< \infty}\big| f(x)\big|. $$
Further, let us consider the following K-functional:
$$ K_{2}(f,\delta)=\inf_{g\in W^{2}} \bigl\{ \| f-g \|+\delta \big\| g^{{\prime\prime}}\big\| \bigr\} , $$
where \(\delta>0\) and \(W^{2}=\{g\in C_{B}[0,\infty):g^{{\prime}},g^{{\prime\prime}}\in C_{B}[0,\infty)\}\). By Theorem 2.4 of [22] there exists an absolute constant \(C>0\) such that
$$ K_{2}(f,\delta)\leq C\omega_{2}(f,\sqrt{\delta}), $$
(3.1)
where
$$ \omega_{2} (f,\sqrt{\delta})=\sup_{0< h\leq\sqrt{\delta}} \sup _{x\in [0,\infty)}\big| f(x+2h)-2f(x+h)+f(x)\big| $$
is the second-order modulus of smoothness of \(f\in C_{B} [0,\infty)\). The usual modulus of continuity of \(f\in C_{B} [0,\infty)\) is defined by
$$ \omega(f,\delta)=\sup_{0< h\leq\delta} \sup_{x\in[0,\infty)}\big| f(x+h)-f(x)\big|. $$

Theorem 3.1

Let \(p,q \in(0,1)\) be such that \(q < p\). Then we have
$$ \big|{S}_{n,p,q}(f;x)-f(x)\big|\leq C \omega_{2} \bigl(f; \delta_{n}(x) \bigr) $$
for every \(x\in[0,\infty)\) and \(f\in C_{B} [0,\infty)\), where
$$ \delta_{n}^{2}(x)=\frac{ x}{[n]_{p,q} }+ \biggl( \frac{1}{p}-1 \biggr)x^{2}. $$

Proof

Let \(g\in\mathcal{W}^{2}\). Then from the Taylor expansion we get
$$ g(t)=g(x)+g^{\prime}(x) (t-x)+ \int_{x}^{t}(t-u)g^{\prime\prime}(u) \,\mathrm{d}u, \quad t\in[0,\mathcal{A}], \mathcal{A}>0. $$
Now by Corollary 2.1 we have
$$\begin{gathered} {S}_{n,p,q}(g;x)=g(x)+{S}_{n,p,q} \biggl( \int_{x}^{t}(t-u)g^{\prime\prime}(u)\,\mathrm{d}u;x \biggr), \\\begin{aligned}{\big|} {S}_{n,p,q}(g;x)-g(x) {\big|} &\leq{S}_{n,p,q} \biggl( {\bigg|}\int_{x}^{t}\big|(t-u)\big| \big| g^{\prime\prime}(u) \big| \mathrm{d}u;x {\bigg|} \biggr) \\ &\leq{S}_{n,p,q} \bigl( (t-x)^{2};x \bigr) \big\| g^{\prime\prime} \big\| . \end{aligned}\end{gathered}$$
Hence we get
$$ {\big|} {S}_{n,p,q}(g;x)-g(x) {\big|}\leq\big\| g^{\prime\prime}\big\| \biggl( \frac{x}{[n]_{p,q}}+ \biggl(\frac{1}{p}-1 \biggr)x^{2} \biggr) . $$
On the other hand, we have
$$ {\big|} {S}_{n,p,q}(f;x)-f(x) {\big|} \leq \big|{S}_{n,p,q} \bigl( (f-g);x \bigr) -(f-g) (x)\big|+ {\big|} {S}_{n,p,q}(g;x)-g(x) {\big|}. $$
Since
$$ \big|{S}_{n,p,q}(f;x)\big|\leq\| f\|, $$
we have
$$ {\big|} {S}_{n,p,q}(f;x)-f(x) {\big|} \leq \| f-g\| +\big\| g^{\prime\prime}\big\| \biggl( \frac{x}{[n]_{p,q}}+ \biggl(\frac {1}{p}-1 \biggr)x^{2} \biggr) . $$
Now taking the infimum on the right-hand side over all \(g\in\mathcal {W}^{2}\), we get
$$ {\big|} {S}_{n,p,q}(f;x)-f(x) {\big|} \leq \mathcal{C}K_{2} \bigl( f, \delta_{n}^{2}(x) \bigr) . $$
By the property of a K-functional we get
$$ {\big|} {S}_{n,p,q}(f;x)-f(x) {\big|} \leq \mathcal{C}\omega _{2} \bigl( f,\delta_{n}(x) \bigr) . $$
This completes the proof. □

4 Weighted approximation by \(S_{n,p,q}\)

Now we give approximation properties of the operators \(S_{n,p,q}\) on the interval \([0,\infty)\). Since
$$\begin{aligned} S_{n,p,q} \bigl(1+t^{2};x \bigr) &=1+ \biggl( \frac{1}{p}-1 \biggr)x^{2}+\frac{x}{[n]_{p,q}} \\ &\leq1+x^{2}+x, \end{aligned}$$
\(x\leq1\) for \(x\in{}[0,1]\), and \(x\leq x^{2}\) for \(x\in(1,\infty)\), we have
$$ S_{n,p,q} \bigl(1+t^{2};x \bigr)\leq2 \bigl(1+x^{2} \bigr), $$
which says that \(S_{n,p,q}\) are linear positive operators acting from \(C_{2}[0,\infty)\) to \(B_{2}[0,\infty)\). For more details, see [23, 24], and [25].

Theorem 4.1

Let the sequence of linear positive operators \((L_{n})\) acting from \(C_{2} [0,\infty)\) to \(B_{2} [0,\infty)\) satisfy the condition
$$ \lim_{n\rightarrow\infty}\| L_{n}e_{i}-e_{i} \|_{2}=0 ,\quad i=0, 1, 2. $$
Then, for any function \(f \in C_{2}^{\ast} [0,\infty) \),
$$ \lim_{n\rightarrow\infty}\| L_{n}f-f\|_{2}=0. $$

Theorem 4.2

Let \(q = q_{n}\in(0, 1)\) and \(p = p_{n}\in(q, 1)\) be such that \(q_{n}\rightarrow1\) and \(p_{n}\rightarrow1\) as \(n \rightarrow\infty\). Then, for each function \(f \in C_{2}^{\ast} [0,\infty)\), we get
$$ \lim_{n\rightarrow\infty}\| S_{n,p_{n},q_{n}}f-f\|_{2}=0. $$

Proof

According to Theorem 4.1, it is sufficient to verify the condition
$$ \lim_{n\rightarrow\infty}\| S_{n,p_{n},q_{n}}e_{i}-e_{i} \|_{2} = 0,\quad i=0, 1, 2. $$
(4.1)
By Lemma 2.1(i), (ii) it is clear that
$$\begin{gathered} \lim_{n\rightarrow\infty}\big\| S_{n,p_{n},q_{n}}(1;x)-1\big\| _{2} =0, \\ \lim_{n\rightarrow\infty}\big\| S_{n,p_{n},q_{n}}(t;x)-x \big\| _{2} =0, \end{gathered}$$
and by Lemma 2.1(iii) we have
$$\begin{aligned} \lim_{n\rightarrow\infty}\big\| S_{n,p_{n},q_{n}} \bigl(t^{2};x \bigr)-x^{2}\big\| _{2} &=\sup_{x\geq0} \frac {(\frac{1}{p_{n}}-1)x^{2}+\frac{x}{[n]_{p_{n},q_{n}}}}{1+x^{2}} \\ &\leq \biggl(\frac{1}{p_{n}}-1 \biggr)+\frac{1}{[n]_{p_{n},q_{n}}}. \end{aligned}$$
The last inequality means that (4.1) holds for \(i=2\). By Theorem 4.1 the proof is complete. □
The weighted modulus of continuity is given by
$$ \Omega(f; \delta) = \sup_{0\leq h < \delta,x\in[0, \infty) } \frac {| f(x+h)-f(x)|}{(1+h^{2})+(1+x^{2})} $$
(4.2)
for \(f \in C_{2} [0,\infty)\). We know that, for every \(f \in C_{2}^{\ast} [0,\infty)\), \(\Omega(\cdot; \delta)\) has the properties
$$ \lim_{\delta\rightarrow0}\Omega(f; \delta)=0 $$
and
$$ \Omega(f; \lambda\delta) \leq 2(1+\lambda) \bigl(1+ \delta^{2} \bigr) \Omega(f; \delta), \quad\lambda>0. $$
(4.3)
For \(f \in C_{2} [0,\infty)\), from (4.2) and (4.3) we can write
$$ \begin{aligned}[b] \big| f(t)-f(x)\big|&\leq \bigl(1+(t-x)^{2} \bigr) \bigl(1+x^{2} \bigr)\Omega\bigl(f; | t-x|\bigr) \\ &\leq 2 \biggl(1+\frac{| t-x|}{\delta} \biggr) \bigl(1+\delta^{2} \bigr) \Omega(f; \delta) \bigl(1+(t-x)^{2} \bigr) \bigl(1+x^{2} \bigr). \end{aligned} $$
(4.4)
All concepts mentioned can be found in [26].

Theorem 4.3

Let \(0< q = q_{n} < p = p_{n}\leq1\) be such that \(q_{n}\rightarrow 1\) and \(p_{n}\rightarrow1\) as \(n \rightarrow\infty\). Then, for each function \(f \in C_{2}^{\ast} [0,\infty)\), there exists a positive constant A such that
$$ \sup_{x\in[0, \infty) } \frac{| S_{n,p,q}(f;x)-f(x)| }{(1+x^{2})^{\frac{5}{2}}} \leq A\Omega \biggl(f; \frac{1}{ \sqrt{\beta_{p,q}(n)}} \biggr), $$
where \(\beta_{p,q}(n)= \max \{\frac{1}{p}-1 ,\frac{1}{[n]_{p,q}} \}\), and A is a positive constant.

Proof

Since \(S_{n,p,q}(1; x) = 1\), using the monotonicity of \(S_{n,p,q}\), we can write
$$ \big| S_{n,p,q}(f;x)-f(x)\big|\leq S_{n,p,q} \bigl(\big| f(t)-f(x) \big|;x \bigr). $$
On the other hand, from (4.4) we have that
$$\begin{aligned} \big| S_{n,p,q}(f;x)-f(x)\big|\leq{}& 2 \bigl(1+\delta^{2} \bigr) \Omega(f; \delta) \bigl(1+x^{2} \bigr) \biggl[S_{n,p,q} \biggl( \biggl(1+\frac{| t-x| }{\delta} \biggr) \bigl(1+(t-x)^{2} \bigr);x \biggr) \biggr] \\ \leq{}& 2 \bigl(1+\delta^{2} \bigr)\Omega(f; \delta) \bigl(1+x^{2} \bigr) \biggl\{ S_{n,p,q}(1;x)+S_{n,p,q} \bigl((t-x)^{2};x \bigr) \\ &+\frac{1}{\delta}S_{n,p,q}\bigl(| t-x|;x \bigr)+\frac{1}{\delta}S_{n,p,q} \bigl(| t-x|(t-x)^{2} ;x \bigr) \biggr\} . \end{aligned}$$
Using the Cauchy-Schwarz inequality, we can write
$$\begin{aligned} \big| S_{n,p,q}(f;x)-f(x)\big|\leq{}& 2 \bigl(1+\delta^{2} \bigr) \Omega(f; \delta) \bigl(1+x^{2} \bigr) \biggl\{ S_{n,p,q}(1;x)+S_{n,p,q} \bigl((t-x)^{2};x \bigr) \\ &+ \frac{1}{\delta} \sqrt{S_{n,p,q} \bigl((t-x)^{2} ;x \bigr)} +\frac{1}{\delta }\sqrt{S_{n,p,q} \bigl( (t-x)^{4} ;x \bigr)} \sqrt{S_{n,p,q} \bigl( (t-x)^{2} ;x \bigr)} \biggr\} . \end{aligned}$$
On the other hand, using (2.3), we have
$$\begin{aligned} S_{n,p,q} \bigl((t-x)^{2} ;x \bigr) &\leq \frac{ x}{[n]_{p,q} }+ \biggl(\frac {1}{p}-1 \biggr)x^{2} \\ &\leq C_{1}O \bigl(\beta_{p,q}(n) \bigr) \bigl(1+x^{2} \bigr), \end{aligned}$$
where \(C_{1} > 0\) and \(\beta_{p,q}(n)= \max \{\frac{1}{p}-1 ,\frac {1}{[n]_{p,q}} \}\). Since \(\lim_{n\rightarrow\infty}\frac {1}{p_{n}}=1\) and \(\lim_{n\rightarrow\infty}\frac{1}{[n]_{p,q}}=0\), there exists a positive constant \(A_{2}\) such that
$$ S_{n,p,q} \bigl((t-x)^{2} ;x \bigr) \leq A_{2} \bigl(1+x^{2} \bigr). $$
Also, using (2.5), we get
$$ S_{n,p,q} \bigl( (t-x)^{4} ;x \bigr)^{\frac{1}{2}} \leq A_{3} \bigl(1+x^{2} \bigr) $$
and
$$ S_{n,p,q} \biggl( \frac{(t-x)^{2}}{\delta^{2}} ;x \biggr)^{\frac{1}{2}} \leq \frac{A_{4}}{\delta} O \bigl(\beta_{p,q}(n) \bigr)^{\frac{1}{2}} \bigl(1+x^{2} \bigr)^{\frac{1}{2}} $$
for \(A_{3} > 0\) and \(A_{4} > 0\). So we have
$$\begin{aligned} \big| S_{n,p,q}(f;x)-f(x)\big|\leq{}& 2 \biggl(1+\frac{1}{\beta_{p,q}(n)} \biggr) \Omega \biggl(f; \frac{1}{\sqrt{\beta_{p,q}(n)}} \biggr) \bigl(1+x^{2} \bigr) \biggl\{ 1+ A_{2} \bigl(1+x^{2} \bigr) \\ &+ \frac{A_{4}}{\delta} O \bigl(\beta_{p,q}(n) \bigr)^{\frac{1}{2}} \bigl(1+x^{2} \bigr)^{\frac{1}{2}} \\ &+A_{3} \bigl(1+x^{2} \bigr)\frac{A_{4}}{\delta}O \bigl(\beta_{p,q}(n) \bigr)^{\frac{1}{2}} \bigl(1+x^{2} \bigr)^{\frac{1}{2}} \biggr\} . \end{aligned}$$
Choosing \(\delta= \beta_{p,q}(n)^{\frac{1}{2}}\), we obtain
$$\begin{aligned} \big| S_{n,p,q}(f;x)-f(x)\big|\leq{}& 2 \bigl(1+\beta_{p,q}(n) \bigr) \Omega \biggl(f; \frac {1}{\sqrt{\beta_{p,q}(n)}} \biggr) \bigl(1+x^{2} \bigr) \bigl\{ 1+ A_{2} \bigl(1+x^{2} \bigr) \\ &+ CA_{4} \bigl(1+x^{2} \bigr)^{\frac{1}{2}} +C_{1}A_{3}A_{4} \bigl(1+x^{2} \bigr)^{\frac {3}{2}} \bigr\} . \end{aligned}$$
For \(0 < q < p \leq1\), we have \(\beta_{p,q}(n) \leq1\). Hence we can write
$$ \sup_{x\in[0, \infty) } \frac{| S_{n,p,q}(f;x)-f(x)| }{(1+x^{2})^{\frac{5}{2}}} \leq A\Omega \biggl(f; \frac{1}{ \sqrt{\beta_{p,q}(n)}} \biggr), $$
where \(A = 4 (1 + A_{2} + CA_{4} + C_{1}A_{3}A_{4})\), and the result follows. □

5 Voronovskaya-type theorem for \(S_{n,p,q}\)

Here we give a Voronovskaya-type theorem for \(S_{n,p,q}\).

Theorem 5.1

Let \(0< q_{n} < p_{n}\leq1\) be such that \(q_{n}\rightarrow 1\), \(p_{n}\rightarrow1\), \(q_{n}^{n}\rightarrow a\), and \(p_{n}^{n}\rightarrow b\) as \(n \rightarrow\infty\). Then, for each function \(f \in C_{2}^{\ast} [0,\infty)\) such that \(f^{{\prime}},f^{{\prime\prime}} \in C_{2}^{\ast} [0,\infty)\), we have
$$ \lim_{n\rightarrow\infty} [n]_{p_{n},q_{n}} \bigl( S_{n,p_{n},q_{n}}(f;x)-f(x) \bigr)= \bigl(x+\alpha x^{2} \bigr)f^{{\prime\prime}}(x) $$
uniformly on any \([0,A] \), \(A > 0\).

Proof

Let \(f,f^{{\prime}},f^{{\prime\prime}} \in C_{2}^{\ast} [0,\infty )\) and \(x \in[0,\infty)\). By the Taylor formula we can write
$$ f(t) = f(x)+f^{{\prime}}(x) (t-x)+\frac{1}{2}f^{{\prime\prime }}(x) (t-x)^{2}+h(t,x) (t-x)^{2}, $$
(5.1)
where \(h (t, x)\) is the remainder of the Peano form. Then \(h (\cdot, x) \in C_{2}^{\ast} [0,\infty)\) and \(\lim_{t\rightarrow x}h (t, x)=0\) for n large enough. Applying operators (2.1) to both sides of (5.1), we get
$$\begin{aligned} [n]_{p_{n},q_{n}} \bigl( S_{n,p_{n},q_{n}}(f;x)-f(x) \bigr)={}&[n]_{p_{n},q_{n}}f^{{\prime }}(x)S_{n,p_{n},q_{n}} \bigl((t-x);x \bigr)\\ & + [n]_{p_{n},q_{n}}f^{{\prime \prime}}(x)S_{n,p_{n},q_{n}} \bigl((t-x)^{2};x \bigr) \\ &+S_{n,p_{n},q_{n}} \bigl(h (t, x) (t-x)^{2};x \bigr). \end{aligned}$$
By the Cauchy-Schwarz inequality we have
$$ S_{n,p_{n},q_{n}} \bigl(h (t, x) (t-x)^{2};x \bigr) \leq \sqrt{S_{n,p_{n},q_{n}} \bigl(h^{2} (t, x);x \bigr)} \sqrt{S_{n,p_{n},q_{n}} \bigl((t-x)^{4};x \bigr)} . $$
(5.2)
Observe that \(h^{2} (x, x) = 0\) and \(h^{2} (\cdot, x)\in C_{2}^{\ast} [0,\infty)\). Then it follows from Theorem 4.3 that
$$ \lim_{n\rightarrow\infty} S_{n,p_{n},q_{n}} \bigl(h^{2} (t, x);x \bigr) = h^{2} (x, x)=0 $$
(5.3)
uniformly with respect to \(x \in[0,A]\). Hence, from (5.2), (5.3), and (2.8) we obtain
$$ \lim_{n\rightarrow\infty}[n]_{p_{n},q_{n}} S_{n,p_{n},q_{n}} \bigl(h (t, x) (t-x)^{2};x \bigr) = 0 $$
(5.4)
and
$$ S_{n,p,q} \bigl((t-x);x \bigr) = 0. $$
Then using (2.6) and (5.4), we have
$$\begin{aligned} \lim_{n\rightarrow\infty} [n]_{p_{n},q_{n}} \bigl( S_{n,p_{n},q_{n}}(f;x)-f(x) \bigr) ={}&f^{{\prime }}(x)\lim_{n\rightarrow\infty} [n]_{p_{n},q_{n}}S_{n,p_{n},q_{n}} \bigl((t-x);x \bigr) \\ &+ f^{{\prime\prime}}(x)\lim_{n\rightarrow\infty} [n]_{p_{n},q_{n}}S_{n,p_{n},q_{n}} \bigl((t-x)^{2};x \bigr) \\ &+ \lim_{n\rightarrow\infty}[n]_{p_{n},q_{n}}S_{n,p_{n},q_{n}} \bigl(h (t, x) (t-x)^{2};x \bigr) \\ ={}& \bigl(x+\alpha x^{2} \bigr)f^{{\prime\prime}}(x), \end{aligned}$$
as desired. □

6 Conclusion

In this paper, we have constructed a new modification of Szász-Mirakyan operators based on \((p,q)\)-integers and investigated their approximation properties. We have obtained a weighted approximation and Voronovskaya-type theorem for our new operators.

Declarations

Acknowledgements

The authors gratefully acknowledge the financial support from King Abdulaziz University, Jeddah, Saudi Arabia.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, Aligarh Muslim University
(2)
Operator Theory and Applications Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University

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