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Chlodowsky variant of q-Bernstein-Schurer-Stancu operators
Journal of Inequalities and Applications volume 2014, Article number: 189 (2014)
It was Chlodowsky who considered non-trivial Bernstein operators, which help to approximate bounded continuous functions on the unbounded domain. In this paper, we introduce the Chlodowsky variant of q-Bernstein-Schurer-Stancu operators. By obtaining the first few moments of these operators, we prove Korovkin-type approximation theorems in different function spaces. Furthermore, we compute the error of the approximation by using the modulus of continuity and Lipschitz-type functionals. Then we obtain the degree of the approximation in terms of the modulus of continuity of the derivative of the function. Finally, we study the generalization of the Chlodowsky variant of q-Bernstein-Schurer-Stancu operators and investigate their approximations.
MSC:41A10, 41A25, 41A36.
The classical Bernstein-Chlodowsky operators were defined by Chlodowsky  as
where the function f is defined on and is a positive increasing sequence with and as .
In 1968, Stancu  constructed and studied the Bernstein-Stancu operators, which were defined as
where such that .
In 1987, Lupaş  introduced the q-based Bernstein operators and obtained the Korovkin-type approximation theorem. In 1996, other q-based Bernstein operators were defined by Phillips [4, 5]. During the last decade q-based operators have become an active research area (see [6–9]).
In 2011, the q-Bernstein-Schurer operators were defined by Muraru , for fixed and for all , by
where q is a positive real number and the function f is evaluated at the q-integers . Recall that the q-integer of is 
the q-factorial is defined by
and q-binomial coefficients are defined by
for , . Note that the case in (1.1), the operators defined by (1.1) were reduced to the operators considered by Schurer .
Some properties of the q-Bernstein-Schurer operators were investigated in . We should notice that the case reduces to the q-Bernstein operators.
It should be noted that complex approximation properties of some Schurer-type operators were investigated in [14, 15] and .
Recently, Barbosu investigated Schurer-Stancu operators which were defined by  (see also )
where α and β are non-negative numbers which satisfy and also p is a non-negative integer.
In 2008, Karslı and Gupta  defined the q-analogue of Chlodowsky operators by
where has the same property of Bernstein-Chlodowsky operators.
Lately, the q-analogues of Bernstein-Schurer-Stancu operators were introduced by Agrawal et al. as 
where α, β and p are non-negative integers such that . For the first few moments, we have the following lemma.
Lemma 1.1 [, p.7755]
For the operator , we have the following moments:
The organization of the paper as follows.
In Section 2, we introduce the Chlodowsky variant of q-Bernstein-Schurer-Stancu operators and investigate the moments of the operator. In Section 3, we study several Korovkin-type theorems in different function spaces. In Section 4, we obtain the order of convergence of the Chlodowsky variant of q-Bernstein-Schurer-Stancu operators by means of Lipschitz class functions and the first modulus of continuity. In addition, we calculate the degree of convergence of the approximation process in terms of the first modulus of continuity of the derivative of the function. In Section 5, we study the generalization of the Chlodowsky variant of q-Bernstein-Schurer-Stancu operators and investigate their approximations.
2 Construction of the operators
We construct the Chlodowsky variant of q-Bernstein-Schurer-Stancu operators as
where and and with , , . Clearly, is a linear and positive operator. Note that the cases and in (2.1) reduce to the Stancu-Chlodowsky polynomials .
Firstly, we obtain the following lemma, which will be used throughout the paper.
Lemma 2.1 Let be given in (2.1). Then the first few moments of the operators are
Proof Using Lemma 1.1 and considering the following facts:
we get the assertions (i), (ii), and (iii).
Using the linearity of the operators, we have
So, we get (iv).
Similar computations give
Then we have
This proves (v). □
Lemma 2.2 For each fixed , we have
Proof Since we get
If we calculate the right-hand side of the above inequality, we get the following:
Remark 2.1 As a consequence of Lemma 2.1 and Lemma 2.2, we have
Lemma 2.3 For the second central moment we have the following estimate:
Proof Taking supremum over in (2.2) we get the result. □
3 Korovkin-type approximation theorem
In this section, we prove Korovkin-type approximation theorem for the Chlodowsky variant of q-Bernstein-Schurer-Stancu operators. Denote by the space of all continuous functions f, satisfying the condition
Obviously, is a linear normed space with the norm
In studying the weighted approximation, the following theorem is crucial.
Theorem 3.1 (See )
There exists a sequence of positive linear operators , acting from to , satisfying the conditions
where is continuous and increasing function on such that and and there exists a function for which .
The following theorem has been given in  and will be used in the investigation of the approximation properties of in weighted spaces.
Theorem 3.2 (See )
The conditions (3.1), (3.2), (3.3) imply for any function f belonging to the subset of for which
Particularly, choosing and applying Theorem 3.2 to the operators
we can state the following theorem.
Theorem 3.3 For all we have
provided that with , and as .
Proof In the proof we directly use Theorem 3.2. Obviously, by Lemma 2.1(i), (ii), and (iii) we get the following inequalities, respectively:
is satisfied since and as . □
Lemma 3.4 Let A be a positive real number independent of n and f be a continuous function which vanishes on . Assume that with , and . Then we have
Proof From the hypothesis on f, one can write (). For arbitrary small , we have
where and are independent of n. With the help of the following equality:
we get by Theorem 3.3 and Remark 2.1
Since as , we have the desired result. □
Theorem 3.5 Let f be a continuous function on the semi-axis and
Assume that with , , , and . Then
Proof The proof will be given along the lines of the proof of Theorem 2.5 in . Clearly, it is sufficient to prove the theorem for the case . Since , given any we can find a point such that
For any fixed , define an auxiliary function as follows:
Then for sufficiently large n in such a way that and in view of , we have
We have from (3.4)
Now, we can write
where for . By Lemma 3.4, we obtain the result. □
4 Order of convergence
In this section, we compute the rate of convergence of the operators in terms of the elements of Lipschitz classes and the modulus of continuity of the function. Additionally, we calculate the order of convergence in terms of the first modulus of continuity of the derivative of the function.
Now, we give the rate of convergence of the operators in terms of the Lipschitz class , for . Let denote the space of bounded continuous functions on . A function belongs to if
Theorem 4.1 Let
Proof Considering the monotonicity and the linearity of the operators, and taking into account that ,
Using Hölder’s inequality with and , we get by the statement (2.2)
Now we give the rate of convergence of the operators by means of the modulus of continuity which is denoted by . Let and . Then the definition of the modulus of continuity of f is given by
It follows that for any the inequality
is satisfied .
Theorem 4.2 If , we have
where is modulus of continuity of f which is defined in (4.1) and be the same as in Theorem 4.1.
Proof By the triangular inequality, we get
Now using (4.2) and the Hölder inequality, we can write
Now choosing the same as in Theorem 4.1, we have
Now, we compute the rate of convergence of the operators in terms of the modulus of continuity of the derivative of the function.
Theorem 4.3 If has a continuous bounded derivative and is the modulus of continuity of in , then
where M is a positive constant such that .
Proof Using the mean value theorem, we have
where ξ is a point between x and . By using the above identity, we get
Using the above inequality, we have
Therefore, applying the Cauchy-Schwarz inequality for the second term, we get
Therefore, using (2.2) we see that
Choosing , we obtain the desired result. □
5 Generalization of the Chlodowsky variant of q-Bernstein-Schurer-Stancu operators
In this section, we introduce generalization of Chlodowsky variant of q-Bernstein-Schurer-Stancu operators. The generalized operators help us to approximate the continuous functions on more general weighted spaces. Note that this kind of generalization was considered earlier for the Bernstein-Chlodowsky polynomials  and q-Bernstein-Chlodowsky polynomials .
For , consider any continuous function and define
Let us take into account the generalization of the as follows:
where and has the same properties as the Chlodowsky variant of the q-Bernstein-Schurer-Stancu operators.
Theorem 5.1 For the continuous functions satisfying
By using and continuity of the function f, we get for and is continuous function on . Thus, from Theorem 3.2 we get the result. □
Finally note that, taking , the operators reduce to .
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The authors declare that they have no competing interests.
All authors completed the paper together. All authors read and approved the final manuscript.
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Vedi, T., Ali Özarslan, M. Chlodowsky variant of q-Bernstein-Schurer-Stancu operators. J Inequal Appl 2014, 189 (2014). https://doi.org/10.1186/1029-242X-2014-189
- Schurer-Stancu and Schurer Chlodowsky operators
- modulus of continuity
- Korovkin-type theorems
- Lipschitz-type functionals
- q-Bernstein operators