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Some statistical approximation properties of Kantorovich-type q-Bernstein-Stancu operators
Journal of Inequalities and Applications volume 2014, Article number: 10 (2014)
Abstract
In this paper two kinds of Kantorovich-type q-Bernstein-Stancu operators are introduced, and the statistical approximation properties of these operators are investigated. Furthermore, by means of modulus of continuity, the rates of statistical convergence of these operators are also studied.
MSC:41A10, 41A25, 41A36.
1 Introduction
In 1997, Phillips [1] introduced and studied q analogue of Bernstein polynomials. During the last decade, the applications of q-calculus in the approximation theory have become one of the main area of research, q-calculus has been extensively used for constructing various generalizations of many classical approximation processes. It is well known that many q-extensions of the classical objects arising in the approximation theory have been recently introduced and studied (e.g., see [2–8]). Recently the statistical approximation properties have also been investigated for q-analogue polynomials. For instance, in [9] Kantorovich-type q-Bernstein operators; in [10]q-Baskakov-Kantorovich operators; in [11] Kantorovich-type q-Szász-Mirakjan operators; in [12]q-Bleimann, Butzer and Hahn operators; in [13]q-analogue of MKZ operators were introduced and their statistical approximation properties were studied.
The goal of this paper is to introduce two kinds of new Kantorovich-type q-Bernstein-Stancu operators and to study the statistical approximation properties of these operators with the help of the Korovkin-type approximation theorem. We also estimate the rate of statistical convergence of the mentioned sequences of operators to the appropriate function f, respectively.
Before proceeding further, let us give some basic definitions and notations from q-calculus. Details on q-integers can be found in [14, 15].
Let , for each nonnegative integer k, the q-integer and the q-factorial are defined by
and
respectively.
Then for and integers n, k, , we have
For the integers n, k, , the q-binomial coefficient is defined by
For an arbitrary function , the q-differential is given by
The q-Jackson integral in the interval is defined as
provided that sums converge absolutely.
Suppose . The q-Jackson integral in a generic interval is defined as
2 Construction of the operators
In this part, we first construct the Kantorovich-type q-Bernstein-Stancu operators as follows.
Definition 1 Let f be a q-integrable function on , for , , , , we define the Kantorovich-type q-Bernstein-Stancu operators by
where
In the following we give some lemmas.
Lemma 1 For , , , we have
Proof For , since , , so Eq. (2) holds.
For ,
For ,
□
Lemma 2 For , , , , we have
Proof In view of Lemma 1 and , by a simple computation, we have
□
Lemma 3 ([9])
Let , , and let f be a positive function defined on the interval . If f is monotone increasing on , then in this interval.
It is clear that the operator is a linear and positive operator for any monotone increasing function .
Remark 1 To guarantee the positivity of , f must be a monotone increasing function on the interval . But for the function f this condition is strong. In order to solve the problems, a special type of q-integral, which is the Riemann-type q-integral, is defined by Marinković et al. [16].
Definition 2 ([16])
Let , . The Riemann-type q-integral is defined as
provided the sums converge absolutely.
We now redefine by putting the Riemann-type q-integral into the operators instead of the general q-integral as
where f is a Riemann-type q-integrable function on .
Let us give some lemmas as follows.
Lemma 4 Let , , then is a linear and positive operator.
Proof The proof is clear, so we omit it. □
Lemma 5 For , , , , we have
Proof By Definition 2, we have
Hence, by using the equality and Eq. (10), we get
By using Eq. (11) and the equality , we have
By using Eq. (12) and the equality , we have
□
Lemma 6 For , , , , we have
Proof In view of Lemma 5, by a simple computation, we have
□
3 Statistical approximation of Korovkin type
Now, let us recall the concept of statistical convergence which was introduced by Fast [17].
Let set and , the natural density of K is defined by if the limit exists (see [18]), 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 .
Note that any convergent sequence is statistically convergent, but not conversely. Details can be found in [19].
In approximation theory, the concept of statistical convergence was used by Gadjiev and Orhan [20]. They proved the following Bohman-Korovkin type approximation theorem for statistical convergence.
Theorem 1 ([20])
If the sequence of linear positive operators satisfies the conditions
for , . Then, for any ,
Theorem 2 Let , , be a sequence satisfying the following condition
then for any monotone increasing function , we have
Proof From Theorem 1, it is enough to prove that for , .
From Eq. (2), we can easily get
From Eq. (3), we have
In view of , for we have
Now, for every given , let us define the following sets:
From inequality (15), one can see that , so we have
By condition (13), it is clear that , .
So we have
In view of Eq. (4) and the equality , by a simple computation, for we have
Now, for every given , let us define the following sets:
It is clear that , so we get
By condition (13), we have
so, we can get
In view of Eqs. (14), (16) and (17), the proof is complete. □
Theorem 3 Let , , be a sequence satisfying condition (13), then for all , we have .
Proof From Theorem 1, it is enough to prove that , for , .
From Eq. (7), we can easily get
From Eq. (8), we have
In view of , for we have
Now, for every given , let us define the following sets:
From inequality (19), one can see that , so we have
By condition (13), it is clear that
So we have
In view of Eq. (9) and the equality , for and , by a simple computation, we have
Now, for every given , let us define the following sets:
It is clear that , so we get
By condition (13), we have
So, we can get
In view of Eqs. (18), (20) and (21), the proof is complete. □
4 Rates of statistical convergence
Let , for any , the usual modulus of continuity for f is defined as .
For and any , we have , so for any , we get
In the light of for , it is clear that we have
for any and .
Next we will give the rates of convergence of both and by means of modulus of continuity.
Theorem 4 Let , be a sequence satisfying condition (13), then for any monotone increasing function , we have
where
Proof Using the linearity and positivity of the operator , by Lemma 2 and inequality (22), we get
Take , , to be a sequence satisfying condition (13) and choose in (23), we have , which implies the proof is complete. □
Theorem 5 Let , , be a sequence satisfying condition (13), then for any , we have , where
Proof Using the linearity and positivity of the operator , by Lemma 6 and inequality (22), we get
Take , , to be a sequence satisfying condition (13) and choose in (24), we have , which implies the proof is complete. □
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Acknowledgements
The authors are most grateful to the editor and anonymous referee for careful reading of the manuscript and valuable suggestions which helped in improving an earlier version of this paper. This work is supported by the National Natural Science Foundation of China (Grant No. 61170324), the Class A Science and Technology Project of Education Department of Fujian Province of China (Grant No. JA12324), and the Natural Science Foundation of Fujian Province of China (Grant No. 2013J01017).
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Ren, MY., Zeng, XM. Some statistical approximation properties of Kantorovich-type q-Bernstein-Stancu operators. J Inequal Appl 2014, 10 (2014). https://doi.org/10.1186/1029-242X-2014-10
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DOI: https://doi.org/10.1186/1029-242X-2014-10