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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)

## Abstract

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.

## 1. Introduction

In 1980, Bleimann et al. [1] 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. [1] 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 [2]. 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 [3], Chêng [4], Guo and Khan [5], Zeng and Piriou [6], Gupta et al. [7], involving several different operators.

Throughout this paper the class of function is defined as follows:

Our main result can be stated as follows.

Theorem 1.1.

Let and let be the total variation of on interval . Then, for sufficiently large, one has

where

## 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].

Lemma 2.1.

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:

where

. It is easy to verify that .

Lemma 2.2.

If is fixed and is sufficiently large, then

(a)for , there holds

(b)for , there holds

Proof.

We first prove (a). Since , , then . Hence, we have

Direct calculation gives

Hence , for *sufficiently large*.

The proof of is similar.

Lemma 2.3 (see [9, Theorem 1] or, cf. [10]).

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

where .

Assuming that , for some (), then we have

Thus

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

where .

Noticing and for , we have .

Thus

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

Furthermore, since

we have

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

where

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 [11], 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).

## References

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**A Bernstein-type operator approximating continuous functions on the semi-axis.***Indagationes Mathematicae*1980,**83**(3):255–262. 10.1016/1385-7258(80)90027-XAbel U, Ivan M:

**A Kantorovich variant of the Bleimann, Butzer and Hahn operators.***Rendiconti del Circolo Matematico di Palermo. Serie II. Supplemento*2002,**68:**205–218.Bojanić R, Vuilleumier M:

**On the rate of convergence of Fourier-Legendre series of functions of bounded variation.***Journal of Approximation Theory*1981,**31**(1):67–79. 10.1016/0021-9045(81)90031-9Chêng F:

**On the rate of convergence of Bernstein polynomials of functions of bounded variation.***Journal of Approximation Theory*1983,**39**(3):259–274. 10.1016/0021-9045(83)90098-9Guo SS, Khan MK:

**On the rate of convergence of some operators on functions of bounded variation.***Journal of Approximation Theory*1989,**58**(1):90–101. 10.1016/0021-9045(89)90011-7Zeng X-M, Piriou A:

**On the rate of convergence of two Bernstein-Bézier type operators for bounded variation functions.***Journal of Approximation Theory*1998,**95**(3):369–387. 10.1006/jath.1997.3227Gupta V, Abel U, Ivan M:

**Rate of convergence of beta operators of second kind for functions with derivatives of bounded variation.***International Journal of Mathematics and Mathematical Sciences*2005,**2005**(23):3827–3833. 10.1155/IJMMS.2005.3827Feller W:

*An Introduction to Probability Theory and Its Applications*.*Volume 2*. 2nd edition. John Wiley & Sons, New York, NY, USA; 1971:xxiv+669.Zeng X-M:

**Bounds for Bernstein basis functions and Meyer-König and Zeller basis functions.***Journal of Mathematical Analysis and Applications*1998,**219**(2):364–376. 10.1006/jmaa.1997.5819Zeng X-M, Zhao J-N:

**Exact bounds for some basis functions of approximation operators.***Journal of Inequalities and Applications*2001,**6**(5):563–575. 10.1155/S1025583401000340Srivastava HM, Gupta V:

**Rate of convergence for the Bézier variant of the Bleimann-Butzer-Hahn operators.***Applied Mathematics Letters*2005,**18**(8):849–857. 10.1016/j.aml.2004.08.014

## Acknowledgments

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

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DOI: https://doi.org/10.1155/2009/852897

### Keywords

- Positive Integer
- Growth Condition
- Total Variation
- Probability Theory
- Limit Theorem