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Statistical approximation of modified Schurer-type q-Bernstein Kantorovich operators
Journal of Inequalities and Applications volume 2014, Article number: 465 (2014)
Abstract
New modified Schurer-type q-Bernstein Kantorovich operators are introduced. The local theorem and statistical Korovkin-type approximation properties of these operators are investigated. Furthermore, the rate of approximation is examined in terms of the modulus of continuity and the elements of Lipschitz class functions.
MSC:41A10, 41A25, 41A36.
1 Introduction
In 1987, Lupaş [1] introduced a q-analogue of Bernstein operators, and in 1997 another q-generalization of the Bernstein polynomials was introduced by Phillips [2]. After that generalizations of the Bernstein polynomials based on the q-integers attracted a lot of interest and were studied widely by a number of authors. Some new generalizations of well-known positive linear operators based on q-integers were introduced and studied by several authors (e.g., see [3–6]). On the other hand, the study of the statistical convergence for sequences of positive operators was attempted by Gadjiev and Orhan [7]. Very recently, the statistical approximation properties have also been investigated for q-analogue polynomials. For instance, in [8]q-Bleimann, Butzer and Hahn operators; in [9] Kantorovich-type q-Bernstein operators; in [10] a q-analogue of MKZ operators; in [11] Kantorovich-type q-Szász-Mirakjan operators; in [12] Kantorovich-type q-Bernstein-Stancu operators were introduced and their statistical approximation properties were studied.
The paper is organized as follows. In Section 2, we introduce a new modification of Schurer-type q-Bernstein Kantorovich operators and evaluate the moments of these operators. In Section 3 we study local convergence properties in terms of the first and the second modulus of continuity. In Section 4, we obtain their statistical approximation properties with the help of the Korovkin-type theorem proved by Gadjiev and Orhan. Furthermore, in Section 5, we compute the degree of convergence of the approximation process in terms of the modulus of continuity and the Lipschitz class functions.
2 Construction of the operators
Some definitions and notations regarding the concept of q-calculus can be found in [5]. Let (the set of all nonnegative integers) be such that . We introduce a new modification of Schurer-type q-Bernstein Kantorovich operators as follows:
where and . It is clear that is a linear and positive operator. When , it reduces to the Schurer-type q-Bernstein Kantorovich operators (see [13])
In order to investigate the approximation properties of , we need the following lemmas.
Lemma 2.1 ([14])
For the generalized q-Schurer-Stancu operators
the following properties hold:
Lemma 2.2 For , , we have
Proof It is obvious that
For , since , so (2.5) holds.
For , we get
Using (2.3), we have
So
For ,
we obtain
Using (2.3) and (2.4), by a simple calculation we can get the stated result (2.7). □
Lemma 2.3 From Lemma 2.2, we have
and
3 Local approximation
Now, we consider a sequence satisfying the following two expressions:
By the Korovkin theorem, we can state the following theorem.
Theorem 3.1 Let be a sequence satisfying (3.1) for . Then, for any function , the following equality
is satisfied.
Proof We know that is linear positive. By Lemma 2.2, if we choose the sequence satisfying conditions (3.1), and using the equality
we have
as . Because of the linearity and positivity of , the proof is complete by the classical Korovkin theorem. □
Consider the following K-functional:
where
Then, in view of a known result [15], there exists an absolute constant such that
where
is the second modulus of smoothness of .
Let , for any , the usual modulus of continuity for f is defined as .
We next present the following local theorem of the operators in terms of the first and the second modulus of continuity of the function .
Theorem 3.2 Let , there exists an absolute constant such that
where
and
Proof Let
where and
Using the Taylor formula
for , we have
Hence
Observe that
we obtain
Using (3.4) and the uniform boundedness of , we get
Taking the infimum on the right-hand side over all , we obtain
which together with (3.3) gives the proof of the theorem. □
4 Korovkin-type statistical approximation properties
Further on, let us recall the concept of statistical convergence which was introduced by Fast [16].
Let the set and , the natural density of K is defined by if the limit exists (see [17]), where denotes the cardinality of the set .
A sequence is called statistically convergent to a number L if for every , . This convergence is denoted as . It is known that any convergent sequence is statistically convergent, but not conversely. Details can be found in [18].
In approximation theory by linear positive operators, the concept of statistical convergence was used by Gadjiev and Orhan [7]. They proved the following Bohman-Korovkin-type approximation theorem for statistical convergence.
Theorem 4.1 ([7])
If the sequence of linear positive operators satisfies the conditions
for , , then, for any ,
In this section, we establish the following Korovkin-type statistical approximation theorems.
Theorem 4.2 Let , , be a sequence satisfying the following conditions:
then, for , we have
Proof From Theorem 4.1, it is enough to prove that for , .
By (2.5), we can easily get
From equality (2.8) and (3.2) we have
Now, for given , let us define the following sets:
From (4.3), one can see that , so we have
By (4.1) it is clear that
and
So we have
Finally, in view of (2.7), one can write
Using (3.2),
and
we can write
Then, from (4.1), we have
Here, for given , let us define the following sets:
It is clear that . So we get
By condition (4.5), we have
which implies that
In view of (4.2), (4.4) and (4.6), the proof is complete. □
5 Rates of convergence
Let for any and . Then we have , so for any , we get
Owing to for , it is obvious that we have
for any , and .
Now, we give the convergence rate of to the function in terms of the modulus of continuity.
Theorem 5.1 Let , , be a sequence satisfying (4.1), then for any function , , we have
where is given by (2.9).
Proof Using the linearity and positivity of the operator and inequality (5.1), for any and , we get
Take , , be a sequence satisfying condition (4.1) and choose in (5.2), the desired result follows immediately. □
Finally, we give the rate of convergence of with the help of functions of the Lipschitz class. We recall a function on if the inequality
holds.
Theorem 5.2 Let on , . Let , be a sequence satisfying the conditions given in (4.1). If we take as in (2.9), then we have
Proof Let on , . Since is linear and positive, by using (5.3), we have
If we take , and apply the Hölder inequality, then we obtain
□
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Acknowledgements
This research is supported by the Fundamental Research Funds for the Central Universities (Nos. N110323010, N130323015), Science and Technology Research Founds for Colleges and Universities in Hebei Province (No. Z2014040), and the Research Fund for Northeastern University at Qinhuangdao (No. XNB201429).
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Lin, Q. Statistical approximation of modified Schurer-type q-Bernstein Kantorovich operators. J Inequal Appl 2014, 465 (2014). https://doi.org/10.1186/1029-242X-2014-465
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DOI: https://doi.org/10.1186/1029-242X-2014-465