- Research Article
- Open Access

# Rate of Convergence of a New Type Kantorovich Variant of Bleimann-Butzer-Hahn Operators

- Lingju Chen
^{1}Email author and - Xiao-Ming Zeng
^{2}Email author

**2009**:852897

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

© L. Chen and X.-M. Zeng. 2009

**Received:**28 September 2009**Accepted:**16 November 2009**Published:**24 November 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.

## Keywords

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

## 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 .

where , , and is Lebesgue measure.

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.

Our main result can be stated as follows.

Theorem 1.1.

## 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.

where , , .

Lemma 2.2.

If is fixed and is sufficiently large, then

Proof.

Hence
, for
*sufficiently large*.

The proof of is similar.

## 3. Proof of Theorem 1.1

Let , and , Bojanic-Cheng decomposition yields

where .

Assuming that , for some ( ), then we have

Set , then by (2.4) and using Lemma 2.1, we have

Finally, we estimate .

where .

Noticing and for , we have .

Next, let .

Now, the remainder of our work is to estimate .

Theorem 1.1 now follows from (3.3), (3.9), and (3.29).

## Declarations

### 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.

## Authors’ Affiliations

## References

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## Copyright

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