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Chlodowsky-type q-Bernstein-Stancu-Kantorovich operators
Journal of Inequalities and Applications volume 2015, Article number: 91 (2015)
Abstract
In this paper, we introduce Chlodowsky-type q-Bernstein-Stancu-Kantorovich operators on the unbounded domain. We should note that this generalization includes various kinds of operators which have not been introduced earlier. We calculate the error of approximation of these operators by using the modulus of continuity and Lipschitz-type functionals. Finally, we give generalization of the operators and investigate their approximations.
1 Introduction
Generalizations of Bernstein polynomials and their q-analogues have been an intensive research area of approximation theory (see [1–19]). In this paper, we introduce the Chlodowsky-type q-Bernstein-Stancu-Kantorovich operators and investigate their approximation properties.
Firstly, let us recall the following notions of q-integers [20]. Let \(q>0\). For any integer \(k\geq0\), the q-integer \([ k ] _{q}= [ k ] \) is defined by
the q-factorial \([ k ] _{q}!= [ k ] !\) is defined by
and for integers \(n\geq k\geq0\), q-binomial coefficients are defined by
In 2011, the q-based Bernstein-Schurer operators were defined by Muraru [21] as
Then she obtained the Korovkin-type theorem and the order of convergence by using the modulus of continuity. She also mentioned that if \(q\rightarrow 1^{-}\) in (1.1), the operators reduce to the Schurer operators considered by Schurer [22] and if \(p=0\) in (1.1), they contain the q-Bernstein operators [16]. After that, different approximation properties of the q-Bernstein-Schurer operators were studied in [23].
Recently, the q-Bernstein-Schurer-Kantorovich operators were defined [24] as
Then, the approximation rates of the q-Bernstein-Schurer-Kantorovich operators were given by means of Lipschitz class functionals and the first and the second modulus of continuity.
Notice that if we choose \(p=0\) in (1.2), we get the q-Bernstein-Kantorovich operators which were defined by Mahmudov and Sabancıgil in [25]. We should also mention that in [26] the authors defined a different version of q-Bernstein-Kantorovich operators, where they used the usual integral instead of q-integral in the definition.
In 2013, the q-analogue of Bernstein-Schurer-Stancu operators \(S_{n,p}^{\alpha,\beta}:C [ 0,1+p ] \rightarrow C [ 0,1 ] \) was introduced by Agrawal et al. in [2] by
where α and β are real numbers which satisfy \(0\leq \alpha \leq\beta\) and also p is a non-negative integer.
Then, Ren and Zeng introduced the Kantorovich-type-q-Bernstein-Stancu operators [27]. They investigated the statistical approximation properties.
On the other hand, Karslı and Gupta [13] introduced q-Chlodowsky operators as follows:
where \(n\in\mathbb{N}\) and \(( b_{n} ) \) is a positive increasing sequence with \(\lim_{n\rightarrow\infty}b_{n}=\infty\). Then, they investigated the approximation properties of \(C_{n,q}(f;x)\).
Recently, the Chlodowsky variant of q-Bernstein-Schurer-Stancu operators was introduced by the authors in [28] as
where \(n\in\mathbb{N}\) and \(p\in\mathbb{N}_{0}\) and \(\alpha, \beta \in \mathbb{R}\) with \(0\leq\alpha\leq\beta\), \(0\leq x\leq b_{n}\), \(0< q<1\), and Korovkin-type approximation theorems were proved in different function spaces. Moreover, the error of approximation was computed by using the modulus of continuity and Lipschitz-type functionals. Also, the generalization of the Chlodowsky variant of q-Bernstein-Schurer-Stancu operators was studied. Notice that \(C_{n,p}^{ ( 0,0 ) } ( f;q;x ) \) gives the q-Bernstein-Schurer-Chlodowsky operators which have not been defined yet, and additionally taking \(p=0\), we get the q-Bernstein-Chlodowsky operators [13]. On the other hand, from [28] the first three moments of \(C_{n,p}^{ ( \alpha,\beta ) }(f;q;x)\) are as follows:
The organization of this paper is as follows. In Section 2, we introduce the Chlodowsky-type q-Bernstein-Stancu-Kantorovich operators and calculate the moments for them. In Section 3, Korovkin-type theorems are proved. In Section 4, we obtain the rate of convergence of the approximation process in terms of the first and the second modulus of continuity and also by means of Lipschitz class functions. In Section 5, we study the generalization of the Kantorovich-Stancu type generalization of q-Bernstein-Chlodowsky operators and study their approximation properties.
2 Construction of the operators
The Chlodowsky-type q-Bernstein-Stancu-Kantorovich operators are introduced as
where \(n\in\mathbb{N}\) and \(p\in\mathbb{N}_{0}\) and \(\alpha, \beta \in \mathbb{R}\) with \(0\leq\alpha\leq\beta\), \(0\leq x\leq b_{n}\), \(0< q<1\). Obviously, \(K_{n,p}^{ ( \alpha,\beta ) }\) is a linear and positive operator. We should notice that if we choose \(p=\alpha=\beta=0\) in (2.1) and taking into account that \(( 1+ ( q-1 ) [ k ] ) t=q^{k}t\), the operator \(K_{n,p}^{ ( \alpha,\beta ) } ( f;q;x ) \) reduces to the Chlodowsky variant of the q-Bernstein Kantorovich operator [26].
First of all let us give the following lemma which will be used throughout the paper.
Lemma 2.1
Let \(K_{n,p}^{ ( \alpha,\beta ) } ( f;q;x ) \) be given in (2.1). Then we have
Proof
(i) Using (2.1) and \(C_{n,q} ( 1;x ) =1\), we get
(ii) After some calculations, we obtain
Whence the result.
(iii) By (2.1) we can write
After some calculations as in (i) and (ii), we get the desired result.
(iv) Using (i) and (ii), we get
(v) It is known that
Then we obtain the result directly. □
Lemma 2.2
For the second central moment, if we take supremum on \([ 0,b_{n} ] \), we get the following estimate:
3 Korovkin-type approximation theorem
In this section, we study Korovkin-type approximation theorems of Chlodowsky-type q-Bernstein-Stancu-Kantorovich operators. Let \(C_{\rho}\) denote the space of all continuous functions f such that the following condition
is satisfied.
It is clear that \(C_{\rho}\) is a linear normed space with the norm
The following theorems play an important role in our investigations.
Theorem 3.1
(See [9])
There exists a sequence of positive linear operators \(U_{n} \), acting from \(C_{\rho}\) to \(C_{\rho}\), satisfying the conditions
where \(\phi ( x ) \) is a continuous and increasing function on \(( -\infty,\infty ) \) such that \(\lim_{x\rightarrow\pm \infty}\phi ( x ) =\pm\infty\) and \(\rho ( x ) =1+\phi ^{2}\), and there exists a function \(f^{\ast}\in C_{\rho}\) for which \(\varlimsup_{n\rightarrow\infty} \Vert U_{n}f^{\ast }- f^{\ast} \Vert _{\rho}>0\).
Theorem 3.2
(See [9])
Conditions (3.1), (3.2), (3.3) imply
for any function f belonging to the subset \(C_{\rho}^{0}:= \{ f\in C_{\rho}:\lim_{\vert x\vert \rightarrow\infty}\frac{f ( x ) }{\rho ( x ) }\textit{ is finite} \} \).
Let us choose \(\rho ( x ) =1+x^{2}\) and consider the operators:
It should be mentioned that the operators \(U_{n}^{(\alpha,\beta)}\) act from \(C_{1+x^{2}}\) to \(C_{1+x^{2}}\). For all \(f\in C_{1+x^{2}}\), we have
Therefore, it is clear from Lemma 2.1 that
provided that \(q:= ( q_{n} ) \) with \(0< q_{n}<1\), \(\lim_{n\rightarrow\infty}q_{n}=1\), \(\lim_{n\rightarrow \infty}q_{n}^{n}=N<\infty\) and \(\lim_{n\rightarrow\infty}\frac{ b_{n}}{ [ n ] }=0\). We have the following approximation theorem.
Theorem 3.3
For all \(f\in C_{1+x^{2}}^{0}\), we have
provided that \(q:= ( q_{n} ) \) with \(0< q_{n}<1\), \(\lim_{n\rightarrow\infty}q_{n}=1\), \(\lim_{n\rightarrow \infty}q_{n}^{n}=N<\infty\) and \(\lim_{n\rightarrow\infty}\frac{b_{n}}{ [ n ] }=0\).
Proof
With help of Theorem 3.2 and Lemma 2.1(i), (ii) and (iii), we have the following estimates, respectively:
and
whenever \(n\rightarrow\infty\), since \(\lim_{n\rightarrow\infty }q_{n}=1\) and \(\frac{b_{n}}{ [ n ] }=0\) as \(n\rightarrow \infty\). □
Lemma 3.4
Let A be a positive real number independent of n and f be a continuous function which vanishes on \([ A,\infty ] \). Assume that \(q:= ( q_{n} ) \) with \(0< q_{n}<1\), \(\lim_{n\rightarrow\infty }q_{n}=1\), \(\lim_{n\rightarrow\infty}q_{n}^{n}=N<\infty\) and \(\lim_{n\rightarrow\infty}\frac{b_{n}^{2}}{ [ n ] }=0\). Then we have
Proof
From the hypothesis on f, one can write \(\vert f(x)\vert \leq M\) (\(M>0\)). For arbitrary small \(\varepsilon>0\), we have
for \(x\in [ 0,b_{n} ] \) and \(\delta=\delta ( \varepsilon ) \). Using the following equality
we get by Lemma 2.2 that
Since \(0< q_{n}<1\), \(\lim_{n\rightarrow\infty}q_{n}=1\), \(\lim_{n\rightarrow\infty}q_{n}^{n}=N<\infty\) and \(\lim_{n\rightarrow\infty}\frac{b_{n}^{2}}{ [ n ] }=0\), we have the desired result. □
Theorem 3.5
Let f be a continuous function on the semi-axis \([0,\infty)\) and
Assume that \(q:= ( q_{n} ) \) with \(0< q_{n}<1\), \(\lim_{n\rightarrow\infty}q_{n}=1\), \(\lim_{n\rightarrow\infty }q_{n}^{n}=K<\infty\) and \(\lim_{n\rightarrow\infty}\frac{b_{n}^{2}}{ [ n ] }=0\). Then
Proof
If we apply the same techniques as in the proof of Theorem 3.5 in [28] and use Lemma 3.4, we obtain the desired result. □
4 Order of convergence
In this section, we study the rate of convergence of the operators in terms of the elements of Lipschitz classes and the first and the second modulus of continuity of the function.
Firstly, we give the rate of convergence of the operators \(K_{n,p}^{ ( \alpha,\beta ) }\) in terms of the Lipschitz class \(\operatorname{Lip}_{M} ( \gamma ) \). Let \(C_{B}[0,\infty)\) denote the space of bounded continuous functions on \([0,\infty)\) endowed with the usual supremum norm. A function \(f\in C_{B}[0,\infty)\) belongs to \(\operatorname{Lip}_{M} ( \gamma ) \) (\(0<\gamma\leq1\)) if the condition
is satisfied.
Theorem 4.1
Let \(f\in \operatorname{Lip}_{M}(\gamma)\). Then we have
where
Proof
Using the monotonicity and the linearity of the operators and taking into account that \(f\in \operatorname{Lip}_{M}(\gamma)\), we get
Applying Hölder’s inequality with \(p=\frac{2}{\gamma}\) and \(q=\frac {2}{2-\gamma}\), we have the following inequalities by (2.2):
Then we get
where \(p_{n,k} ( q;x ) =\sum_{k=0}^{n+p} \bigl[\scriptsize{\begin{array}{@{}c@{}} n+p \\ k \end{array} }\bigr] ( \frac{x}{b_{n}} ) ^{k}\prod_{s=0}^{n+p-k-1} ( 1-q^{s}\frac{x}{b_{n}} ) \). Again using Hölder’s inequality with \(p=\frac{2}{\gamma}\) and \(q=\frac{2}{2-\gamma}\), we have
where \(\delta_{n,q} ( x ) :=K_{n,p}^{(\alpha,\beta)} ( ( t-x ) ^{2};q;x ) \). □
Now, we give the rate of convergence of the operators by means of the modulus of continuity \(\omega(f;\delta)\). Let \(f\in C_{B}[0,\infty)\) such that f is uniformly continuous and \(x\geq0\). The modulus of continuity of f is given as
It is known that for any \(\delta>0\) the following inequality
is satisfied [8].
Theorem 4.2
If \(f\in C_{B} [ 0,\infty ) \), we have
where \(\delta_{n,q} ( x ) \) is the same as in Theorem 4.1.
Proof
From monotonicity, we have
Now by (4.2) we get
Then, using the Cauchy-Schwarz inequality, we have
Finally, let us choose \(\delta_{n,q}(x) \) the same as in Theorem 4.1. Then we get
 □
Now let us denote by \(C_{B}^{2} [ 0,\infty ) \) the space of all functions \(f\in C_{B} [ 0,\infty ) \) such that \(f^{\prime },f^{\prime\prime}\in C_{B} [ 0,\infty ) \). Let \(\Vert f\Vert \) denote the usual supremum norm of f. The classical Peetre’s K-functional and the second modulus of smoothness of the function \(f\in C_{B} [ 0,\infty ) \) are defined respectively as
and
where \(\delta>0\). It is known that (see [8], p.177) there exists a constant \(A>0\) such that
Theorem 4.3
Let \(q\in ( 0,1 ) \), \(x\in [ 0,b_{n} ] \) and \(f\in C_{B} [ 0,\infty ) \). Then, for fixed \(p\in\mathbb{N}_{0}\), we have
for some positive constant C, where
and
Proof
Define an auxiliary operator \(K_{n,p}^{\ast} ( f;q;x ) :C_{B} [ 0,\infty ) \rightarrow C_{B} [ 0,\infty ) \) by
Then by Lemma 2.1 we get
For given \(g\in C_{B}^{2} [ 0,\infty ) \), it follows by the Taylor formula that
Taking into account (4.5) and using (4.7), we get
Then by (4.6)
Since
and
we get
Hence Lemma 2.1 implies that
Since \(\|K_{n,p}^{\ast} ( f;q;\cdot ) \|\leq3\Vert f\Vert \), considering (4.4) and (4.5), for all \(f\in C_{B} [ 0,\infty ) \) and \(g\in C_{B}^{2} [ 0,\infty ) \), we may write from (4.8) that
which yields that
where
and
Hence we get the result. □
5 Generalization of the Kantorovich-Stancu type generalization of q-Bernstein-Chlodowsky operators
In this section, we introduce a generalization of Chlodowsky-type q-Bernstein-Stancu-Kantorovich operators. For \(x\geq0\), consider any continuous function \(\omega ( x ) \geq1\) and define
Let us consider the generalization of \(K_{n,p}^{ ( \alpha,\beta ) } ( f;q;x ) \) as follows:
where \(0\leq x\leq b_{n}\) and \(( b_{n} ) \) has the same properties of Chlodowsky variant of q-Bernstein-Schurer-Stancu-Kantorovich operators.
Notice that this kind of generalization was considered earlier for the Bernstein-Chlodowsky polynomials [9], q-Bernstein-Chlodowsky polynomials [5] and Chlodowsky variant of q-Bernstein-Schurer-Stancu operators [28].
Now we have the following approximation theorem.
Theorem 5.1
For the continuous functions satisfying
we have
provided that \(q:= ( q_{n} ) \) with \(0< q_{n}<1\), \(\lim_{n\rightarrow\infty}q_{n}=1\) and \(\lim_{n\rightarrow \infty}\frac{b_{n}}{ [ n ] }=0\) as \(n\rightarrow\infty\).
Proof
Obviously,
hence
From \(\vert f ( x ) \vert \leq M_{f}\omega ( x ) \) and the continuity of the function f, we have \(\vert G_{f} ( x ) \vert \leq M_{f} ( 1+x^{2} ) \) for \(x\geq0\) and \(G_{f} ( x ) \) is a continuous function on \([ 0,\infty ) \). Using Theorem 3.3, we get the desired result. □
Finally note that taking \(\omega(x)=1+x^{2}\), the operator \(L_{n,p}^{\alpha,\beta} ( f;q;x ) \) reduces to \(K_{n,p}^{(\alpha ,\beta)} ( f;q;x ) \).
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Vedi, T., Özarslan, M.A. Chlodowsky-type q-Bernstein-Stancu-Kantorovich operators. J Inequal Appl 2015, 91 (2015). https://doi.org/10.1186/s13660-015-0610-y
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DOI: https://doi.org/10.1186/s13660-015-0610-y