Abstract
We introduce a new modification of \((p,q)\)-analogue of Szász–Mirakjan operators. Firstly, we give a recurrence relation for the moments of \((p,q)\)-analogue of Szász–Mirakjan operators and present some explicit formulae for the moments and central moments up to order 4. Next, we obtain quantitative estimates for the convergence in the polynomial weighted spaces. In addition, we give the Voronovskaya theorem for the new \((p,q)\)-Szász–Mirakjan operators.
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1 Introduction
In the last two decades, quantum calculus plays a significant role in the approximation of functions by a positive linear operator. In 1987, Lupaş [1] introduced the Bernstein (rational) polynomials based on the q-integers. In 1996, Phillips [2] introduced another generalization of Bernstein polynomials based on q-integers. In [3–23], in the case of \(0< q<1\), many operators have been introduced and examined. Among the most important operators there are q-Szász operators. In [18–21] the authors constructed and studied different q-generalizations of Szász–Mirakjan operators in the case \(0< q<1\). In 2012, Mahmudov [24] introduced the q-Szász operator in the case \(q>1\) and studied quantitative estimates of convergence in polynomial weighted spaces and gave the Voronovskaya theorem.
In recent years, the rapid rise of \((p,q)\)-calculus has led to the discovery of new generalizations of Bernstein polynomials containing \((p,q)\)-integers. In 2015, Mursaleen [25] introduced \((p,q)\)-Bernstein operators and studied approximation properties based on a Korovkin-type approximation theorem of \((p,q)\)-Bernstein operators. Also, In 2017, Khan and Lobiyal [26] constructed a \((p,q)\)-analogue of Lupaş-Bernstein functions. In [27–39], the authors constructed many operators by using \((p,q)\)-integers and studied their approximation properties. Acar [40] introduced \((p,q)\)-Szász–Mirakjan operators. In addition, Acar gave a recurrence relation for the moments of these operators. In the same year, H. Sharma and C. Gupta [41] introduced the generalization of the \((p,q)\)-Szász–Mirakjan Kantorovich operators and examined their approximation properties. In 2017, Mursaleen, AAH Al-Abied, and Alotaibi [42] constructed new Szász–Mirakjan operators based on \((p,q)\)-calculus and studied weighted approximation and a Voronovskaya-type theorem. Also, \((p,q)\)-analogues of Szász–Mirakjan–Baskakov operators [18] and Stancu-type Szász–Mirakjan–Baskakov operators [43] were defined, and their approximation properties were investigated. Acar, Agrawal, and Kumar [44] introduced a sequence of \((p,q)\)-Szász–Mirakjan operators, and their weighted approximation properties were investigated.
2 Construction of \(K_{l,p,q}\) and moment estimations
We give some basic notations and definitions of the \((p,q)\)-calculus.
The \((p,q)\)-integer and \(( p,q ) \)-factorial are defined by
For integers \(0\leq k\leq l\), the \((p,q)\)-binomial is defined by
The \(( p,q ) \)-derivative \(D_{p,q}g\) of a function \(g(z)\) is defined by
The product and quotient formulae for the \((p,q)\)-derivative are as follows:
It is known that
The \((p,q)\)-analogues of an exponential function, denoted by \(e_{p,q} ( z ) \) and \(E_{p,q} ( z ) \), are defined by
and the \((p,q)\)-derivatives of \(e_{p,q} ( az ) \) and \(E_{p,q} ( z ) \) are
Further, the \((p,q)\)-power is defined by
For any integer l,
and \(D_{p,q} ( z-y ) _{p,q}^{0}=0\).
The formula of the kth \((p,q)\)-derivative of the polynomial \(( z-y ) _{p,q}^{l} \)is
where \(l\in \mathbb{Z} _{+}\) and \(0\leq k\leq l\).
The \((p,q)\)-analogue of the Taylor formulas for any function \(g(z)\) is defined by
Let \(C_{\beta}\) denote the set of all real-valued continuous functions g on \([ 0,\infty ) \) such that \(w_{\beta}g\) is bounded and uniformly continuous on \([ 0,\infty ) \) endowed with the norm
where \(w_{0}(z)=1\), and \(w_{\beta}(z)=\frac{1}{1+z^{\beta}}\) for \(\beta \in \mathbb{N}\).
The corresponding Lipschitz class is given for \(0<\alpha\leq2\) by
Now we introduce the \(( p,q ) \)-Szász–Mirakjan operator.
Definition 1
Let \(0< q< p\leq1\) and \(l\in\mathbb{N}\). For \(g: [ 0,\infty ) \rightarrow R\), we define the \(( p,q )\)-Szász–Mirakjan operator as
It is clear that the operator \(K_{l,p,q}\) is linear and positive. It is known that the moments \(K_{l,p,q} ( t^{m};z ) \) play a fundamental role in the approximation theory of positive operators.
Lemma 2
Let\(0< q< p\leq1\)and\(m\in \mathbb{N} \). We have the following recurrence formula:
Proof
According to the definition of \(K_{l,p,q}\) (8), we have
Next, we use the identity \(q [ k ] _{p,q}+p^{k}= [ k+1 ] _{p,q}\) to get the desired formula
□
Lemma 3
Let\(0< q< p\leq1\), \(z\in [ 0,\infty ) \), \(l\in l\), and\(k\geq0\). We have the following identities related to the\(( p,q ) \)-derivative:
where\(s_{lk} ( p,q;z ) =\frac{p^{k(k-l)}}{q^{k ( k-1 ) /2}}\frac{ [ l ] _{p,q}^{k} z^{k}}{ [ k ] _{p,q}!}e_{p,q} ( - [ l ] _{p,q} p^{k-l+1}q^{-k}z ) \).
Proof
We take the \((p,q)\)-derivative of \(s_{lk} ( p,q;z ) \):
Then
Using the obtained formula and the definition of the operator \(K_{l,p,q}\), we get the second desired formula:
□
Lemma 4
For\(0< q< p\leq1\)and\(l\in \mathbb{N} \), we have
Proof
From the \((p,q)\)-Taylor theorem [45] we have
For \(t=0\), having in mind the equalities
for \(\psi_{l} ( z ) =e_{p,q} ( - [ l ] _{p,q} p^{-(l-1)}z ) \), we get the formula
that is, \(K_{l,p,q} ( 1;z ) =1\).
For \(i=2,3,4\), recurrence formula (10) gives us the following results:
□
Lemma 5
For every\(z\in [ 0,\infty ) \), we have
Proof
In fact, we may easily calculate third- and fourth-order central moments as follows:
□
Remark 6
For \(0< q< p \leq1\),
In our study, we assume that \(q=q_{l}\in ( 0,1 ) \) and \(p=p_{l}\in ( q,1 ] \) are such that
and
Therefore
For all \(0< q< p\leq1\) and \(j\geq0\), the \(( p,q ) \)-difference operators are defined as
and
where \(z_{j}=\frac{p^{l-j} [ j ] _{p,q}}{ [ l ] _{p,q}}\). Using this definition, we can prove the following lemmas.
Lemma 7
For all\(0< q< p\leq1\)and\(j,k\in \mathbb{N} \cup \{ 0 \}\), we have
Lemma 8
For all\(0< q< p\leq1\), we have
Lemma 9
We have
Lemma 10
The\((p,q)\)-Szász–Mirakjan operator can be represented as
Proof
Indeed,
□
The next result gives an explicit formula for the moments \(K_{l,p,q} ( t^{m};z ) \) in terms of Stirling numbers, which is a \((p,q)\)-analogue of Becker’s formula; see [46].
Lemma 11
For\(0< q< p\leq1\)and\(m\in l\), we have
where
are the second-type Stirling polynomials satisfying the equalities
Clearly, \(K_{l,p,q} ( t^{m};z ) \)are polynomials of degreemwithout a constant term.
Proof
Because of \(K_{l,p,q} ( t;z ) =z\) and \(K_{l,p,q} ( t^{2};z ) =z^{2}+\frac{z}{p^{-(l-1)} [ l ] _{p,q}}\), representation (13) holds for \(m=1,2\) with \(\mathbb{S}_{p,q} ( 2,1 ) =1\), \(\mathbb{S}_{p,q} ( 1,1 ) =1\).
Using mathematical induction, assume (13) to be valued for m. Then from Lemma 3 we get
□
Remark 12
For \(p=q=1\), formulae (14) become recurrence formulas satisfied by the second-type Stirling numbers from [8].
3 \(( p,q ) \)-Szász–Mirakjan operators in a polynomial weighted space
Lemma 13
For given any fixed\(\beta\in\mathbb{N}\cup \{ 0 \} \)and\(0< q< p\leq1\), we have
where\(K_{1} ( p,q,\beta ) \)are positive constants. Moreover, for every\(g\in C_{\beta}\), we have
Thus\(K_{l,p,q}\)is a linear positive operator from\(C_{\beta}\)into\(C_{\beta}\).
Proof
Inequality (15) is obvious for \(\beta=0\). Let \(\beta\geq1\). Then by (13) we have
where \(K_{1} ( p,q,\beta ) >0\) is a constant depending on β, p, and q. From this (15) follows. Moreover, for every \(g\in C_{\beta}\),
By applying (15) we obtain
□
Lemma 14
For given any fixed\(\beta\in\mathbb{N}\cup \{ 0 \} \)and\(0< q< p\leq1\), we have
where\(K_{2} ( p,q,\beta ) \)are positive constants.
Proof
Formula (11) imply (17) for \(\beta=0\). We have
for \(\beta,l\in\mathbb{N}\). If \(\beta=1\), then we get
which by Lemma 5 yields (17) for \(\beta=1\).
Let \(\beta\geq2\). By applying (13) we get
where \(\mathcal{\wp}_{\beta} ( p,q;z ) \) is a polynomial of degree β. Therefore we have
□
In the next theorem, we give an approximation property of \(K_{l,p,q}\).
Theorem 15
Let\(g\in C_{p}^{2}\), \(0< q< p\leq1\), and\(z\in{}[ 0,\infty)\). There exist positive constants\(K_{3} ( p,q,\beta ) >0\)such that
Proof
By the Taylor formula
we obtain that
□
We consider the modified Steklov means
which have the following properties:
and therefore
We may prove the following so-called direct approximation theorem.
Theorem 16
For given any\(\beta\in\mathbb{N\cup} \{ 0 \} ,g\in C_{\beta}\), \(z\in{}[0,\infty)\), and\(0< q< p\leq1\), we have
Particularly, if\(\mathrm{Lip}_{\beta}^{2}\alpha\)for some\(\alpha\in(0,2]\), then
Proof
For \(g\in C_{\beta}\) and \(h>0\),
and therefore
Since \(w_{\beta}(z)K_{l,p,q} ( \frac{1}{w_{\beta}(t)};z ) \leq K_{1} ( p,q,\beta ) \),we get that
Thus choosing \(h=\sqrt{\frac{z}{p^{-(l-1)} [ l ] _{p,q}}}\), we complete the proof. □
Corollary 17
If\(\beta\in \mathbb{N} \cup \{ 0 \} \), \(g\in C_{\beta}\), \(0< q< p\leq1\), and\(z\in {}[0,\infty)\), then
uniformly on every\([ c,d ] \), \(0\leq c< d\).
4 Convergence of \(( p,q ) \)-Szász–Mirakjan operators
In [47, Theorem 1] and [48, Theorem1], Totik and de la Cal investigated the class problem of all continuous functions g such that \(K_{l,p,q} ( g ) \) converges to g uniformly on the whole interval \([0,\infty)\) as \(l\rightarrow\infty\). The following thorem is a \(( p,q ) \)-analogue of Theorem 1 in [48].
Theorem 18
Assume that\(g: [ 0,\infty ) \rightarrow \mathbb{R} \)is either bounded or uniformly continuous. Let
Then, for all\(t>0\)and\(z\geq0\),
Therefore\(K_{l,p,q} ( g;z ) \)converges toguniformly on\([ 0,\infty ) \)as\(l\rightarrow\infty\)whenever\(g^{\ast}\)is uniformly continuous.
Proof
By the definition of \(g^{\ast}\) we have
Thus we can write
Finally, from the inequality
we obtain
To complete the proof, we need to show that for all \(t>0\) and \(z>0\), we have
Indeed, from the Cauchy–Schwarz inequality it follows that
□
Our next results is a Voronovskaya-type theorem for \(( p,q ) \)-Szász–Mirakjan operators.
Theorem 19
Let\(0< q< p\leq1\). For any\(g\in C_{\beta}^{2} [ 0,\infty ) \), we have the equality
for every\(z\in [ 0,\infty ) \).
Proof
Let \(z\in [ 0,\infty ) \) be fixed. By the Taylor formula we may write
where \(r ( t;z ) \) is the Peano form of the remainder, \(r ( \cdot;z ) \in C_{\beta}\), and \(\lim_{t\rightarrow z}r ( t;z ) =0\). Applying \(K_{l,p,q}\) to (19), we obtain
Applying the Cauchy–Schwarz inequality, we have
Obviously, \(r^{2} ( z;z ) =0\). Then it follows from Corollary 17 that
Now from (20), (21), and Lemma 5 we immediately get
The proof is completed. □
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Kara, M., Mahmudov, N.I. \((p,q)\)-Generalization of Szász–Mirakjan operators and their approximation properties. J Inequal Appl 2020, 116 (2020). https://doi.org/10.1186/s13660-020-02390-0
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DOI: https://doi.org/10.1186/s13660-020-02390-0