Some statistical approximation properties of Kantorovich-type q-Bernstein-Stancu operators
© Ren and Zeng; licensee Springer. 2014
Received: 9 August 2013
Accepted: 9 December 2013
Published: 6 January 2014
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.
In 1997, Phillips  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  Kantorovich-type q-Bernstein operators; in q-Baskakov-Kantorovich operators; in  Kantorovich-type q-Szász-Mirakjan operators; in q-Bleimann, Butzer and Hahn operators; in 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.
provided that sums converge absolutely.
2 Construction of the operators
In this part, we first construct the Kantorovich-type q-Bernstein-Stancu operators as follows.
In the following we give some lemmas.
Proof For , since , , so Eq. (2) holds.
Lemma 3 ()
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. .
Definition 2 ()
provided the sums converge absolutely.
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. □
3 Statistical approximation of Korovkin type
Now, let us recall the concept of statistical convergence which was introduced by Fast .
Let set and , the natural density of K is defined by if the limit exists (see ), 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 .
In approximation theory, the concept of statistical convergence was used by Gadjiev and Orhan . They proved the following Bohman-Korovkin type approximation theorem for statistical convergence.
Theorem 1 ()
Proof From Theorem 1, it is enough to prove that for , .
By condition (13), it is clear that , .
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 , .
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 any and .
Next we will give the rates of convergence of both and by means of modulus of continuity.
Take , , to be a sequence satisfying condition (13) and choose in (23), we have , which implies the proof is complete. □
Take , , to be a sequence satisfying condition (13) and choose in (24), we have , which implies the proof is complete. □
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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