Abstract
The concern of this paper is to introduce a Kantorovich modification of \(( p,q ) \)-Baskakov operators and investigate their approximation behaviors. We first define a new \(( p,q ) \)-integral and construct the operators. The rate of convergence in terms of modulus of continuities, quantitative and qualitative results in weighted spaces, and finally pointwise convergence of the operators for the functions belonging to the Lipschitz class are discussed.
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1 Introduction
The \(( p,q ) \)-calculus is a generalization of the well-known q-calculus and it is constructed by the following notations and definitions. Let \(0< q< p\leq1\). For each nonnegative integer n, the \((p,q)\)-number is denoted by \([ n ] _{p,q}\) and is given by
For each \(k,n\in\mathbb{N}\), \(n\geq k\geq0\), the \((p,q)\)-factorial \([k ] _{p,q}!\) and \((p,q)\)-binomial are defined by
The \((p,q)\)-power basis is defined by
and
Let \(f:\mathbb{R\rightarrow R}\) then 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 following assertions hold true:
and \(D_{p,q}(x\oplus a)_{p,q}^{0}=0\). The formula for the \((p,q)\)-derivative of a product is
Let \(f:C [ 0,a ] \rightarrow \mathbb{R}\) (\(a>0\)) then the \((p,q)\)-integration of a function f is defined by
The formula of the \((p,q)\)-integration by parts is given by
Here we note that all the notations mentioned above reduce to the q-analogs when \(p=1\). For more details of the \((p,q)\)-calculus, we refer the reader to [1–5].
The \((p,q)\)-calculus has been used efficiently in many fields of science such as oscillator algebra, Lie group, field theory, differential equations, hypergeometric series, physical sciences. Therefore, to approximate the functions via polynomials based on \((p,q)\)-integers, no doubt, would have a crucial role. To fulfill this necessity, very recently the well-known sequences of linear positive operators of approximation theory have been transferred to the \((p,q)\)-calculus and the advantages of \((p,q)\) analogs of them have been intensively investigated. For some recent work devoted to \((p,q)\)-operators, we refer the reader to [6–11]. Very recently, Aral and Gupta [12] introduced the \(( p,q ) \)-analog of the well-known Baskakov operators by
where \(x\in{}[0,\infty)\), \(0< q< p\leq1\), and
and they calculated that
Another problem in the approximation by linear positive operators is to present an approximation process for Riemann integrable functions. The main tool to solve this problem is to consider the Kantorovich modifications of the corresponding operators, which mainly depends on the replacing the sample values \(f ( k/n ) \) by the mean values of f in the intervals \([ k/ ( n+1 ) , ( k+1 ) / ( n+1 ) ] \). Since the \(( p,q ) \)-integral of f over \([ a,b ] \) is defined as follows:
one cannot say (1.5) is positive every time unless it is assumed that f is a nondecreasing function. Hence, use of (1.5) to introduce a Kantorovich modification of any \(( p,q ) \)-operators may lead to some problem. Recently Mursaleen et al. [13] introduced a Kantorovich modification of \(( p,q ) \)-Szász-Mirakjan operators using the \(( p,q ) \)-integral (1.5) for the functions being nondecreasing. However, in this paper we define a new \(( p,q ) \)-integral, hence we do not need to impose any condition on f. For the generalizations of Baskakov operators and Kantorovich operators in classical calculus and q-calculus, we refer the reader to some recent papers [14–19].
The aim of this paper is to introduce \(( p,q ) \)-Baskakov-Kantorovich operators and investigate their approximation properties. In the next section, we construct the operators, calculate the moments, central moments of the operators, and give some lemmas which will be necessary to prove our main results. In Section 3, we prove a local approximation theorem for the new operators in terms of Peetre’s \(\mathcal{K}\)-functional and its equivalent modulus of continuities. In Section 4, we investigate the uniform convergence of the operators and present the rate of convergence via the weighted modulus of continuities. In the last section, we give some pointwise estimates for the functions belonging to Lipschitz space.
2 Construction of operators
To present a method to solve the problem mentioned in the Introduction now we propose a new definition of the \(( p,q ) \)-integral. Let \(a,b\in \mathbb{R}\), \(a< b\), and \(h ( x ) :=f ( a+x ) \) be an arbitrary function and \(D_{p,q}H ( x ) =h ( x ) \), where \(H ( x ) :=F ( a+x ) \), then we can write
that is, \(H ( px ) -H ( qx ) = ( p-q ) xh ( x ) \). Hence we get
which allows us to write
Adding these formulas term by term, we have
and taking the limit as \(n\rightarrow\infty\) with the fact \(\vert \frac{p}{q}\vert <1\) we have
Similarly we have, for \(\vert \frac{q}{p}\vert <1\),
and if we take \(x=b-a\) then we get
Definition 1
Let f be an arbitrary function. The \(( p,q ) \)-integral of f can be defined by
Considering the new \(( p,q ) \)-integral given in (2.1) we can define the Kantorovich modifications of the operators (1.3) as follows.
Definition 2
For \(x\in{}[0,\infty)\), \(0< q< p\leq1\), the \(( p,q ) \)-analog of the Baskakov-Kantorovich operators is defined as
Lemma 1
For \(0< q< p\leq1\) and \(n\in\mathbb{N}\) we have
Proof
The proof easily follows from (2.1). □
Lemma 2
For \(x\in{}[0,\infty)\), \(0< q< p\leq1\), \(n\in \mathbb{N}\), the following hold:
where \(e_{i} ( x ) =x^{i}\), \(i=0,1,2\).
Proof
By the definition of the operators (2.2) and equality (2.3) we obtain \(B_{n,p,q}^{\ast} ( e_{0};x ) =B_{n,p,q} ( 1;x ) =1\). In a similar way, using (2.4) we can write
Using the equalities \(B_{n,p,q} ( e_{0};x ) =1\), \(B_{n,p,q} ( e_{1};x ) =x\) we immediately have
Finally, using (2.5) we have
And the equalities (1.4) give us
which completes the proof. □
Remark 1
Using Lemma 2, we get
where
Further, choosing
we can write
Remark 2
For \(q\in ( 0,1 ) \) and \(p\in ( q,1 ] \) we easily see that \(\lim_{n\rightarrow\infty} [ n ] _{p,q}=1/ ( p-q ) \). Hence, the operators (2.2) are not approximation processes with the above form. In order to study the convergence properties of the sequence of \((p,q)\)-Baskakov-Durrmeyer operators, we assume that \(q= ( q_{n} ) \) and \(p= ( p_{n} ) \) 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 \(n\rightarrow\infty\).
Here we note that with these assumptions \(\alpha_{1} ( n ) \rightarrow0\), \(\alpha_{2} ( n ) \rightarrow0\), \(\alpha_{3} ( n ) \rightarrow0\) as \(n\rightarrow\infty\), hence \(\alpha^{\ast } ( n ) \rightarrow0\) as \(n\rightarrow\infty\).
Let \(C_{B} [ 0,\infty ) \) denote the space of all real valued continuous and bounded functions on \([ 0,\infty ) \). In this space we consider the norm
Lemma 3
Let \(f\in C_{B} [ 0,\infty ) \). Then for all \(g\in C_{B}^{2} [ 0,\infty ) \), we have
where \(\tilde{B}_{n,p,q}^{\ast}\) is an auxiliary operator defined by
and
Proof
By the definition of \(\tilde{B}_{n,p,q}^{\ast}\) and Lemma 2, it is obvious that
Since \(g\in C_{B}^{2} [ 0,\infty ) \), using the Taylor expansion for \(x\in [ 0,\infty ) \) we have
Applying the operators \(\tilde{B}_{n,p,q}^{\ast}\) to both sides of the above equality and considering the fact (2.12) we obtain
Also we get
and
Using the inequalities (2.14) and (2.15) in (2.13) we immediately have
□
3 Local approximation
Let us consider the following \(\mathcal{K}\) functional:
where \(\delta>0\) and \(W^{2}= \{ g\in C_{B} [ 0,\infty ) :g^{\prime},g^{\prime\prime}\in C_{B} [ 0,\infty ) \} \). By [20], p.177, Theorem 2.4, 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
Let us recall the definitions of the weighted spaces and corresponding modulus of continuity. Let \(C [ 0,\infty ) \) be the set of all continuous functions f defined on \([ 0,\infty ) \) and \(B_{2} [ 0,\infty ) \) the set of all functions f defined on \([ 0,\infty ) \) satisfying the condition \(\vert f ( x ) \vert \leq M ( 1+x^{2} ) \) with some positive constant M which may depend only on f. \(C_{2} [ 0,\infty ) \) denotes the subspace of all continuous functions in \(B_{2} [ 0,\infty ) \). By \(C_{2}^{\ast} [ 0,\infty ) \), we denote the subspace of all functions \(f\in C_{2} [ 0,\infty ) \) for which \(\lim_{x\rightarrow \infty}\frac{f ( x ) }{1+x^{2}}\) is finite. \(B_{2} [ 0,\infty ) \) is a linear normed space with the norm \(\Vert f\Vert _{2}=\sup_{x\geq0}\frac{\vert f ( x ) \vert }{1+x^{2}}\).
Theorem 1
Let \(f\in C_{B} [ 0,\infty ) \). Then for every \(x\in [ 0,\infty ) \), there exists a constant \(L>0\) such that
Proof
By (2.11), for every \(g\in C_{B}^{2} [ 0,\infty ) \) one can obtain
Taking into account (2.2), (2.6), and (2.11) we have
Using this inequality and Lemma 3 we get
and taking the infimum on the right-hand side over all \(g\in C_{B}^{2} [ 0,\infty ) \) and using (3.1), we deduce
where \(L=4M>0\). □
Theorem 2
Let \(f\in C_{2} [ 0,\infty ) \), \(p_{n},q_{n}\in ( 0,1 ) \) such that \(0< q_{n}< p_{n}\leq1\) and \(\omega_{a+1} ( f,\delta ) \) be its modulus of continuity on the finite interval \([ 0,a+1 ] \subset [ 0,\infty ) \), where \(a>0\). Then the inequality
holds, where \(M_{f}\) is positive constant independent of n and \(\alpha ^{\ast} ( n ) \) is as indicated in (2.9).
Proof
By [21], \(\omega_{a+1} ( \cdot,\delta ) \) has the property
Applying the Cauchy-Schwarz inequality and choosing \(\delta=\sqrt {\alpha^{\ast } ( n ) ( 1+x ) ^{2}}\), we have
which completes the proof. □
4 Weighted approximation
Theorem 3
Let \(q=q_{n}\in ( 0,1 ) \), \(p=p_{n}\in ( q,1 ] \) such that \(q_{n}\rightarrow1\), \(p_{n}\rightarrow1\) as \(n\rightarrow\infty\). Then for each function \(f\in C_{2}^{\ast} [ 0,\infty ) \) we get
Proof
According to the weighted Korovkin theorem proved in [22], it is sufficient to verify the following three conditions:
By (2.6), (4.1) holds for \(i=0\). By (2.7) and (2.8) we have
and by a similar consideration we have
The last two inequalities mean that (4.1) holds for \(i=1,2\). Hence, the proof is completed. □
To obtain the rate of convergence, we consider the weighted modulus of continuity defined by
for \(f\in C_{x^{2}}^{\ast} [ 0,\infty ) \), and \(\Omega _{2} (\cdot;\cdot ) \) has the following properties.
Lemma 4
([23])
If \(f\in C_{x^{2}}^{\ast} [ 0, \infty ) \) then
-
(i)
\(\Omega_{2} ( f;\delta ) \) is monotone increasing function of δ,
-
(ii)
\(\lim_{\delta\rightarrow0^{+}}\Omega_{2} ( f;\delta ) =0\),
-
(iii)
for any \(\lambda\in [ 0,\infty ) \), \(\Omega_{2} ( f;\lambda\delta ) \leq ( 1+\lambda ) \Omega _{2} ( f;\delta ) \).
Theorem 4
Let \(p=p_{n}\) and \(q=q_{n}\) satisfy \(0< q_{n}< p_{n}\leq1\) and for n sufficiently large \(p_{n}\rightarrow 1\), \(q_{n}\rightarrow1\), and \(q_{n}^{n}\rightarrow1\) and \(p_{n}^{n}\rightarrow 1\). If \(f\in C_{x^{2}}^{\ast} [ 0,\infty ) \), then for sufficiently large n we have
where \(\lambda\geq1\) and K is a positive constant independent of f and n, \(\alpha^{\ast} ( n ) \) is as indicated in (2.9).
Proof
By the definition of the weighted modulus of continuity and Lemma 4, we can write
The above inequality allows us to write
Using the Cauchy-Schwarz inequality we have
On the other hand, by (2.8) we get
for sufficiently large n, where \(K_{1}\) is a positive constant. From (4.3), there exists \(K_{2}>0\) such that \(B_{n,p_{n},q_{n}}^{\ast } ( 1+ ( 2x+t ) ^{2};x ) \leq K_{2} ( 1+x^{2} ) \), for sufficiently large n. In a similar way we get
where \(K_{3}\) is a positive constant. Hence we have \(\sqrt{B_{n,p_{n},q_{n}}^{\ast} ( ( 1+ ( 2x+t ) ^{2} ) ^{2};x ) }\leq K_{4} ( 1+x^{2} ) \), for sufficiently large n. Hence we have
Hence choosing \(\delta_{n}=\sqrt{\alpha^{\ast} ( n ) }\) we have
for sufficiently large n and \(x\in [ 0,\infty ) \), where \(K:=K_{2}+K_{4}\). □
Corollary 1
With the assumptions of Theorem 4, if we take the limit as \(n\rightarrow\infty\) in (4.2) we have
5 Pointwise estimates
Theorem 5
Let \(0<\alpha\leq1\) and E be any subset of the interval \([ 0,\infty ) \). Then, if \(f\in C_{B} [ 0,\infty ) \) is locally in \(\operatorname{Lip} ( \alpha ) \), i.e., the condition
holds, then, for each \(x\in [ 0,\infty ) \), we have
where L is a constant depending on α and f; and \(d ( x,E ) \) is the distance between x and E defined by
Proof
Let Ē denote the closure of E in \([ 0,\infty ) \). Then there exists a point \(x_{0}\in\bar{E}\) such that \(\vert x-x_{0}\vert =d ( x,E ) \). Using the triangle inequality
we immediately have by (5.1)
Using the Hölder inequality with \(p=2/\alpha\), \(q=2/ ( 2-\alpha ) \), we obtain
□
Next we obtain the local direct estimate for the operators \(B_{n,p,q}^{\ast } \), using the Lipschitz type maximal function of order α introduced by Lenze [24]:
Theorem 6
Let \(f\in C_{B} [ 0,\infty ) \) and \(0<\alpha\leq1\). Then, for all \(x\in [ 0,\infty ) \) we have
Proof
From equation (5.2), we have
Applying the Hölder inequality with \(p=2/\alpha\), \(q=2/ ( 2-\alpha ) \), we get
□
Remark 3
The further properties of the operators such as convergence properties via summability methods (see, for example, [25, 26]) can be studied.
Conclusion 1
To introduce Kantorovich modifications of the approximation operators in \(( p,q ) \)-calculus, the existing \(( p,q ) \)-integral did not meet the purposes since the positivity of the operators was not guaranteed. In this paper, we solved this problem and presented a new Riemann type \(( p,q ) \)-integral. As an application, we introduced the \(( p,q ) \)-Baskakov-Kantorovich operators and investigated their approximation properties. Using the new \(( p,q ) \)-integral, one can introduce Kantorovich modifications of other well-known operators.
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Acknowledgements
This article was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR for technical and financial support.
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Acar, T., Aral, A. & Mohiuddine, S.A. On Kantorovich modification of \(( p,q ) \)-Baskakov operators. J Inequal Appl 2016, 98 (2016). https://doi.org/10.1186/s13660-016-1045-9
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DOI: https://doi.org/10.1186/s13660-016-1045-9