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.
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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 [7–17], 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
and
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
and
Also the \((p,q)\)-derivative of a function f, denoted by \(D_{p,q}f\), is defined by
provided that f is differentiable at 0. The formula for the \((p,q)\)-derivative of a product is
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:
and
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
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
Proof
Using the identity
we can write
as desired. □
Lemma 2.2
Let \(0< q< p\leq1\) and \(n\in\mathbb{N}\). We have
-
(i)
\({S}_{n,p,q}(1;x)=1\),
-
(ii)
\({S}_{n,p,q}(t;x)=x\),
-
(iii)
\({S}_{n,p,q}(t^{2};x)= \frac{x^{2}}{p}+\frac{ x}{[n]_{p,q} } \),
-
(iv)
\({S}_{n,p,q}(t^{3};x)= \frac{x^{3}}{p^{3}}+ \frac{ 2p+q}{p^{2}[n]_{p,q} }x^{2} + \frac{ x}{[n]_{p,q}^{2} }\),
-
(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
-
(iii)
$$\begin{aligned} {S}_{n,p,q} \bigl(t^{2};x \bigr) ={}& \sum _{k=0}^{\infty} \frac{ p^{ \frac{k(k-1)}{2}}}{ q^{ \frac{k(k-1)}{2}}}\frac{([n]_{p,q}x)^{k}}{ [k]_{p,q}! } \frac {[k]_{p,q}^{2}}{p^{2k-2}[n]_{p,q}^{2}}e_{p,q} \bigl(-[n]_{p,q}q ^{-k}x \bigr) \\ ={}& \sum_{k=0}^{\infty} \frac{ p^{k}p^{ \frac{k(k-1)}{2}}}{ q^{k}q^{ \frac{k(k-1)}{2}}} \frac{([n]_{p,q}x)^{k}}{ [k]_{p,q}! } \frac{[k+1]_{p,q}x}{p^{2k}[n]_{p,q}}e_{p,q} \bigl(-[n]_{p,q}q ^{-(k+1)}x \bigr) \\ ={}& \sum_{k=0}^{\infty}\frac{ p^{k}p^{ \frac{k(k-1)}{2}}}{ q^{k}q^{ \frac{k(k-1)}{2}}} \frac{([n]_{p,q}x)^{k}}{ [k]_{p,q}! }\frac{p^{k}x}{p^{2k}[n]_{p,q}}e_{p,q} \bigl(-[n]_{p,q}q ^{-(k+1)}x \bigr) \\ &+ \sum_{k=0}^{\infty}\frac{ p^{ \frac{k(k-1)}{2}}}{ q^{k}q^{ \frac{k(k-1)}{2}}} \frac{([n]_{p,q}x)^{k}}{ [k]_{p,q}! }\frac{q [k]_{p,q}x }{p^{k}[n]_{p,q}}e_{p,q} \bigl(-[n]_{p,q}q ^{-(k+1)}x \bigr) \\ ={}& \frac{ x}{[n]_{p,q} }+ \sum_{k=0}^{\infty} \frac{ p^{ \frac{k(k-1)}{2}}}{ q^{2k}q^{ \frac {k(k-1)}{2}}}\frac{([n]_{p,q}x)^{k}}{ [k]_{p,q}! }\frac{x^{2} }{p} e_{p,q} \bigl(-[n]_{p,q}q ^{-(k+2)}x \bigr) \\ ={}& \frac{x^{2}}{p}+\frac{ x}{[n]_{p,q} }. \end{aligned}$$
-
(iv)
$$\begin{aligned} {S}_{n,p,q} \bigl(t^{3};x \bigr) ={}& \sum _{k=0}^{\infty} \frac{ p^{ \frac{k(k-1)}{2}}}{ q^{ \frac{k(k-1)}{2}}}\frac{([n]_{p,q}x)^{k}}{ [k]_{p,q}! } \frac {[k]_{p,q}^{3}}{p^{3k-3}[n]_{p,q}^{3}}e_{p,q} \bigl(-[n]_{p,q}q ^{-k}x \bigr) \\ ={}& \sum_{k=0}^{\infty} \frac{ p^{ \frac{k(k-1)}{2}}}{ q^{k}q^{ \frac{k(k-1)}{2}}} \frac{([n]_{p,q}x)^{k}}{ [k]_{p,q}! } \frac {(p^{2k}+2p^{k}q[k]_{p,q}+q^{2}[k]_{p,q}^{2})}{p^{2k}[n]_{p,q}^{2}}\\ &\times x e_{p,q} \bigl(-[n]_{p,q}q ^{-(k+1)}x \bigr) \\ ={}& \sum_{k=0}^{\infty} \frac{ p^{ \frac{k(k-1)}{2}}}{q^{k} q^{ \frac{k(k-1)}{2}}} \frac{([n]_{p,q}x)^{k}}{ [k]_{p,q}! } \frac{x}{[n]_{p,q}^{2}}e_{p,q} \bigl(-[n]_{p,q}q ^{-(k+1)}x \bigr) \\ & +\sum_{k=0}^{\infty} \frac{ p^{ \frac{k(k-1)}{2}}}{ q^{k}q^{ \frac{k(k-1)}{2}}} \frac{([n]_{p,q}x)^{k}}{ [k]_{p,q}! } \frac{2q[k]_{p,q}}{p^{k}[n]_{p,q}^{2}}x e_{p,q} \bigl(-[n]_{p,q}q ^{-(k+1)}x \bigr) \\ &+ \sum_{k=0}^{\infty} \frac{ p^{ \frac{k(k-1)}{2}}}{ q^{k}q^{ \frac{k(k-1)}{2}}} \frac{([n]_{p,q}x)^{k}}{ [k]_{p,q}! } \frac{q^{2}[k]_{p,q}^{2}}{p^{2k}[n]_{p,q}^{2}}x e_{p,q} \bigl(-[n]_{p,q}q ^{-(k+1)}x \bigr) \\ ={}& \frac{ x}{[n]_{p,q}^{2} } +\frac{ 2x^{2}}{p[n]_{p,q} }\\ &+ \sum_{k=0}^{\infty} \frac{p^{k} p^{ \frac{k(k-1)}{2}}}{ q^{2k}q^{ \frac{k(k-1)}{2}}}\frac{([n]_{p,q}x)^{k}}{ [k]_{p,q}! } \frac{q x^{2} (p^{k}+ q[k]_{p,q})}{p^{2k+2}[n]_{p,q}} e_{p,q} \bigl(-[n]_{p,q}q ^{-(k+2)}x \bigr) \\ ={}& \frac{ x}{[n]_{p,q}^{2} } +\frac{ 2x^{2}}{p[n]_{p,q} }+ \frac{ qx^{2}}{p^{2}[n]_{p,q} } \\ &+ \sum _{k=0}^{\infty} \frac{ p^{ \frac{k(k-1)}{2}}}{ q^{2k}q^{ \frac{k(k-1)}{2}}}\frac{([n]_{p,q}x)^{k}}{ [k]_{p,q}! } \frac{q^{2} x^{2} [k]_{p,q}}{p^{k+2}[n]_{p,q}} e_{p,q} \bigl(-[n]_{p,q}q ^{-(k+2)}x \bigr) \\ ={}& \frac{x^{3}}{p^{3}}+ \frac{ 2p+q}{p^{2}[n]_{p,q} }x^{2} + \frac{ x}{[n]_{p,q}^{2} }. \end{aligned}$$
-
(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:
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
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
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
Further, let us consider the following K-functional:
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
where
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
Theorem 3.1
Let \(p,q \in(0,1)\) be such that \(q < p\). Then we have
for every \(x\in[0,\infty)\) and \(f\in C_{B} [0,\infty)\), where
Proof
Let \(g\in\mathcal{W}^{2}\). Then from the Taylor expansion we get
Now by Corollary 2.1 we have
Hence we get
On the other hand, we have
Since
we have
Now taking the infimum on the right-hand side over all \(g\in\mathcal {W}^{2}\), we get
By the property of a K-functional we get
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
\(x\leq1\) for \(x\in{}[0,1]\), and \(x\leq x^{2}\) for \(x\in(1,\infty)\), we have
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
Then, for any function \(f \in C_{2}^{\ast} [0,\infty) \),
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
Proof
According to Theorem 4.1, it is sufficient to verify the condition
By Lemma 2.1(i), (ii) it is clear that
and by Lemma 2.1(iii) we have
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
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
and
For \(f \in C_{2} [0,\infty)\), from (4.2) and (4.3) we can write
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
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
On the other hand, from (4.4) we have that
Using the Cauchy-Schwarz inequality, we can write
On the other hand, using (2.3), we have
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
Also, using (2.5), we get
and
for \(A_{3} > 0\) and \(A_{4} > 0\). So we have
Choosing \(\delta= \beta_{p,q}(n)^{\frac{1}{2}}\), we obtain
For \(0 < q < p \leq1\), we have \(\beta_{p,q}(n) \leq1\). Hence we can write
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
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
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
By the Cauchy-Schwarz inequality we have
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
uniformly with respect to \(x \in[0,A]\). Hence, from (5.2), (5.3), and (2.8) we obtain
and
Then using (2.6) and (5.4), we have
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.
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The authors gratefully acknowledge the financial support from King Abdulaziz University, Jeddah, Saudi Arabia.
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Mursaleen, M., Al-Abied, A. & Alotaibi, A. On \((p,q)\)-Szász-Mirakyan operators and their approximation properties. J Inequal Appl 2017, 196 (2017). https://doi.org/10.1186/s13660-017-1467-z
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DOI: https://doi.org/10.1186/s13660-017-1467-z