Chlodowsky variant of q-Bernstein-Schurer-Stancu operators
© Vedi and Ali Özarslan; licensee Springer. 2014
Received: 14 October 2013
Accepted: 20 January 2014
Published: 13 May 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.
where the function f is defined on and is a positive increasing sequence with and 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]).
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.
where α and β are non-negative numbers which satisfy and also p is a non-negative integer.
where has the same property of Bernstein-Chlodowsky operators.
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]
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
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.
we get the assertions (i), (ii), and (iii).
So, we get (iv).
This proves (v). □
Proof Taking supremum over in (2.2) we get the result. □
3 Korovkin-type approximation theorem
In studying the weighted approximation, the following theorem is crucial.
Theorem 3.1 (See )
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 )
we can state the following theorem.
provided that with , and as .
is satisfied since and as . □
Since as , we have the desired result. □
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.
is satisfied .
where is modulus of continuity of f which is defined in (4.1) and be the same as in Theorem 4.1.
Now, we compute the rate of convergence of the operators in terms of the modulus of continuity of the derivative of the function.
where M is a positive constant such 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 .
where and has the same properties as the Chlodowsky variant of the q-Bernstein-Schurer-Stancu operators.
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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