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Rate of Convergence of a New Type Kantorovich Variant of Bleimann-Butzer-Hahn Operators
Journal of Inequalities and Applications volume 2009, Article number: 852897 (2009)
A new type Kantorovich variant of Bleimann-Butzer-Hahn operator is introduced. Furthermore, the approximation properties of the operators are studied. An estimate on the rate of convergence of the operators for functions of bounded variation is obtained.
In 1980, Bleimann et al.  introduced a sequence of positive linear Bernstein-type operators (abbreviated in the following by BBH operators) defined on the infinite interval by
where denotes the set of natural numbers.
Bleimann et al.  proved that as for (the space of all bounded continuous functions on ) and give an estimate on the rate of convergence of measured with the second modulus of continuity of .
In the present paper, we introduce a new type of Kantorovich variant of BBH operator , also defined on by
where , , and is Lebesgue measure.
The operator (1.2) is different from another type of Kantorovich variant of BBH operator :
which was first considered by Abel and Ivan in . The integrand function in the operator (1.3) has been replaced with new integrand function in the operator (1.2). In this paper we will study the approximation properties of for the functions of bounded variation. The rate of convergence for functions of bounded variation was investigated by many authors such as Bojanić and Vuilleumier , Chêng , Guo and Khan , Zeng and Piriou , Gupta et al. , involving several different operators.
Throughout this paper the class of function is defined as follows:
Our main result can be stated as follows.
Let and let be the total variation of on interval . Then, for sufficiently large, one has
2. Some Lemmas
In order to prove Theorem 1.1, we need the following lemmas for preparation. Lemma 2.1 is the well-known Berry-Esséen bound for the classical central limit theorem of probability theory. Its proof and further discussion can be founded in Feller [8, page 515].
Let be a sequence of independent and identically distributed random variables. And , , then, there holds
where , , .
In addition, let be the random variables with two-point distribution
where . Then we can easily obtain that
Let , then we also have
On the other hand, can be written by following integral form:
. It is easy to verify that .
If is fixed and is sufficiently large, then
(a)for , there holds
(b)for , there holds
We first prove (a). Since , , then . Hence, we have
Direct calculation gives
Hence , for sufficiently large.
The proof of is similar.
For every , there holds
3. Proof of Theorem 1.1
Let , and , Bojanic-Cheng decomposition yields
where is defined as in (1.6) and
Obviously, . Thus it follows from (3.1) that
First of all, we estimate
Assuming that , for some (), then we have
By Lemma 2.3 combining some direct computations, we can easily obtain
Set , then by (2.4) and using Lemma 2.1, we have
Thus, by (3.7), (3.8) we have
Finally, we estimate .
First, interval can be decomposed into four parts as
So can be divided into four parts
Noticing and for , we have .
Next, let .
Now, we recall the Lebesgue-Stieltjes integral representation, and by using partial Lebesgue-Stieltjes integration, we get
An application of (a) in Lemma 2.2 yields
Putting in the last integral, we have
It follows from (3.16) and (3.17) that
By a similar method and using Lemma 2.2(b), we obtain
Now, the remainder of our work is to estimate .
For satisfying the growth condition for some positive integer as , we obviously have
Thus, for sufficiently large, there exists a , such that the following inequalities hold:
where . By the definition of the Stirling numbers of the second kind, we readily have
where the Stirling numbers satisfy
Thus we can write
From , , we can easily find . For a fixed , when , we have . Thus there holds
Now using the similar method as that in the proof of Lemma 4 of , we deduce that
From (3.21), (3.24), and (3.27), we obtain
Finally, by combining (3.12), (3.18), (3.19), and (3.28), we deduce that
Theorem 1.1 now follows from (3.3), (3.9), and (3.29).
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This work is supported by the National Natural Science Foundation of China and the Fujian Provincial Science Foundation of China. The authors thank the associate editor and the referee(s) for their several important comments and suggestions which improve the quality of the paper.
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Chen, L., Zeng, XM. Rate of Convergence of a New Type Kantorovich Variant of Bleimann-Butzer-Hahn Operators. J Inequal Appl 2009, 852897 (2009). https://doi.org/10.1155/2009/852897
- Positive Integer
- Growth Condition
- Total Variation
- Probability Theory
- Limit Theorem