© Özarslan and Vedi; licensee Springer. 2013
Received: 19 November 2012
Accepted: 9 July 2013
Published: 1 October 2013
In the present paper, we introduce the q-Bernstein-Schurer-Kantorovich operators. We give the Korovkin-type approximation theorem and obtain the rate of convergence of this approximation by means of the first and the second modulus of continuity. Moreover, we compute the order of convergence of the operators in terms of the elements of Lipschitz class functions and the modulus of continuity of the derivative of the function.
MSC: 41A10, 41A25, 41A36.
In 1987, q-based Bernstein operators were defined and studied by Lupaş . In 1996, another q-based Bernstein operator was proposed by Phillips . Then the q-based operators have become an active research area (see [5–9] and ).
where . If we choose in (1.1), we get the classical q-Bernstein operators .
Afterwards, several properties and results of the operators defined by (1.1), such as the order of convergence of these operators by means of Lipschitz class functions, the first and the second modulus of continuity and the rate of convergence of the approximation process in terms of the first modulus of continuity of the derivative of the function, were given by the authors . On the other hand, q-Szasz-Schurer operators were discussed in .
Then she obtained the first three moments and gave the rate of convergence of the approximation process in terms of the first modulus of continuity .
So, throughout this paper, we will use the following results, which are computed directly by the tools of q-calculus.
Recall that the first three moments of the q-Bernstein-Schurer operators were given by Muraru in  as follows.
We organize the paper as follows.
Firstly, in section two, we define the q-Bernstein-Schurer-Kantorovich operators and obtain the moments of them. In section three, we obtain the rate of convergence of the q-Bernstein-Schurer-Kantorovich operators in terms of the first modulus of continuity. Also we give the order of approximation by means of Lipschitz class functions and the first and the second modulus of continuity. Furthermore, we compute the degree of convergence of the approximation process in terms of the first modulus of continuity of the derivative of the function.
2 Construction of the operators
for any real number , and . It is clear that is a linear and positive operator for .
For the first three moments and the first and the second central moment, we state the following lemma.
- (ii)Using (1.1), (1.4) and Lemma 1.1, we have
- (iii)From (1.1), (1.4), (1.5) and then Lemma 1.1, we can calculate the as follows:
- (iv)It is obvious that
- (v)Direct calculations yield,(2.2)
By Korovkin’s theorem, we can state the following theorem. □
provided that with and that .
3 Rate of convergence
In this section, we compute the rate of convergence of the operators in terms of the modulus of continuity, elements of Lipschitz classes and the first and the second modulus of continuity of the function. Furthermore, we calculate the rate of convergence in terms of the first modulus of continuity of the derivative of the function.
where is the modulus of continuity of f and , which is given as Lemma 2.1.
The proof is concluded. □
where is the same as in Theorem 3.1.
Hence, the desired result is obtained. □
Hence we get the result. □
Now, we compute the rate of convergence of the operators in terms of the modulus of continuity of the derivative of the function.
This completes the proof. □
4 Concluding remarks
In this paper, we obtain many results in the pointwise sense. On the other hand, we see that the interval is bounded and closed, and also f is continuous on it, so these results can be given in the uniform sense.
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