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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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F852897/MediaObjects/13660_2009_Article_2019_Equ1_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F852897/MediaObjects/13660_2009_Article_2019_Equ2_HTML.gif)
where ,
, and
is Lebesgue measure.
The operator (1.2) is different from another type of Kantorovich variant of BBH operator :
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F852897/MediaObjects/13660_2009_Article_2019_Equ3_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F852897/MediaObjects/13660_2009_Article_2019_Equ4_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F852897/MediaObjects/13660_2009_Article_2019_Equ5_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F852897/MediaObjects/13660_2009_Article_2019_Equ6_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F852897/MediaObjects/13660_2009_Article_2019_Equ7_HTML.gif)
where ,
,
.
In addition, let be the random variables with two-point distribution
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F852897/MediaObjects/13660_2009_Article_2019_Equ8_HTML.gif)
where . Then we can easily obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F852897/MediaObjects/13660_2009_Article_2019_Equ9_HTML.gif)
Let , then we also have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F852897/MediaObjects/13660_2009_Article_2019_Equ10_HTML.gif)
On the other hand, can be written by following integral form:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F852897/MediaObjects/13660_2009_Article_2019_Equ11_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F852897/MediaObjects/13660_2009_Article_2019_Equ12_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F852897/MediaObjects/13660_2009_Article_2019_IEq38_HTML.gif)
. It is easy to verify that .
Lemma 2.2.
If is fixed and
is sufficiently large, then
(a)for , there holds
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F852897/MediaObjects/13660_2009_Article_2019_Equ13_HTML.gif)
(b)for , there holds
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F852897/MediaObjects/13660_2009_Article_2019_Equ14_HTML.gif)
Proof.
We first prove (a). Since ,
, then
. Hence, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F852897/MediaObjects/13660_2009_Article_2019_Equ15_HTML.gif)
Direct calculation gives
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F852897/MediaObjects/13660_2009_Article_2019_Equ16_HTML.gif)
Hence , for
sufficiently large.
The proof of is similar.
Lemma 2.3 (see [9, Theorem  1] or, cf. [10]).
For every , there holds
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F852897/MediaObjects/13660_2009_Article_2019_Equ17_HTML.gif)
3. Proof of Theorem 1.1
Let , and
, Bojanic-Cheng decomposition yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F852897/MediaObjects/13660_2009_Article_2019_Equ18_HTML.gif)
where is defined as in (1.6) and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F852897/MediaObjects/13660_2009_Article_2019_Equ19_HTML.gif)
Obviously, . Thus it follows from (3.1) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F852897/MediaObjects/13660_2009_Article_2019_Equ20_HTML.gif)
First of all, we estimate
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F852897/MediaObjects/13660_2009_Article_2019_Equ21_HTML.gif)
where .
Assuming that , for some
(
), then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F852897/MediaObjects/13660_2009_Article_2019_Equ22_HTML.gif)
Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F852897/MediaObjects/13660_2009_Article_2019_Equ23_HTML.gif)
By Lemma 2.3 combining some direct computations, we can easily obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F852897/MediaObjects/13660_2009_Article_2019_Equ24_HTML.gif)
Set , then by (2.4) and using Lemma 2.1, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F852897/MediaObjects/13660_2009_Article_2019_Equ25_HTML.gif)
Thus, by (3.7), (3.8) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F852897/MediaObjects/13660_2009_Article_2019_Equ26_HTML.gif)
Finally, we estimate .
First, interval can be decomposed into four parts as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F852897/MediaObjects/13660_2009_Article_2019_Equ27_HTML.gif)
So can be divided into four parts
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F852897/MediaObjects/13660_2009_Article_2019_Equ28_HTML.gif)
where .
Noticing and for
, we have
.
Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F852897/MediaObjects/13660_2009_Article_2019_Equ29_HTML.gif)
Next, let .
Now, we recall the Lebesgue-Stieltjes integral representation, and by using partial Lebesgue-Stieltjes integration, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F852897/MediaObjects/13660_2009_Article_2019_Equ30_HTML.gif)
An application of (a) in Lemma 2.2 yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F852897/MediaObjects/13660_2009_Article_2019_Equ31_HTML.gif)
Furthermore, since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F852897/MediaObjects/13660_2009_Article_2019_Equ32_HTML.gif)
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F852897/MediaObjects/13660_2009_Article_2019_Equ33_HTML.gif)
Putting in the last integral, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F852897/MediaObjects/13660_2009_Article_2019_Equ34_HTML.gif)
It follows from (3.16) and (3.17) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F852897/MediaObjects/13660_2009_Article_2019_Equ35_HTML.gif)
By a similar method and using Lemma 2.2(b), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F852897/MediaObjects/13660_2009_Article_2019_Equ36_HTML.gif)
Now, the remainder of our work is to estimate .
For satisfying the growth condition
for some positive integer
as
, we obviously have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F852897/MediaObjects/13660_2009_Article_2019_Equ37_HTML.gif)
Thus, for sufficiently large, there exists a
, such that the following inequalities hold:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F852897/MediaObjects/13660_2009_Article_2019_Equ38_HTML.gif)
where . By the definition of the Stirling numbers
of the second kind, we readily have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F852897/MediaObjects/13660_2009_Article_2019_Equ39_HTML.gif)
where the Stirling numbers satisfy
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F852897/MediaObjects/13660_2009_Article_2019_Equ40_HTML.gif)
Thus we can write
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F852897/MediaObjects/13660_2009_Article_2019_Equ41_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F852897/MediaObjects/13660_2009_Article_2019_Equ42_HTML.gif)
From ,
, we can easily find
. For a fixed
, when
, we have
. Thus there holds
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F852897/MediaObjects/13660_2009_Article_2019_Equ43_HTML.gif)
Now using the similar method as that in the proof of Lemma  4 of [11], we deduce that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F852897/MediaObjects/13660_2009_Article_2019_Equ44_HTML.gif)
From (3.21), (3.24), and (3.27), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F852897/MediaObjects/13660_2009_Article_2019_Equ45_HTML.gif)
Finally, by combining (3.12), (3.18), (3.19), and (3.28), we deduce that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F852897/MediaObjects/13660_2009_Article_2019_Equ46_HTML.gif)
Theorem 1.1 now follows from (3.3), (3.9), and (3.29).
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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