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

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

(1.1)

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

(1.2)

where , , and is Lebesgue measure.

The operator (1.2) is different from another type of Kantorovich variant of BBH operator :

(1.3)

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:

(1.4)

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

(1.5)

where

(1.6)

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

(2.1)

where , , .

In addition, let be the random variables with two-point distribution

(2.2)

where . Then we can easily obtain that

(2.3)

Let , then we also have

(2.4)

On the other hand, can be written by following integral form:

(2.5)

where

(2.6)

. It is easy to verify that .

Lemma 2.2.

If is fixed and is sufficiently large, then

(a)for , there holds

(2.7)

(b)for , there holds

(2.8)

Proof.

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

(2.9)

Direct calculation gives

(2.10)

Hence , for sufficiently large.

The proof of is similar.

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

For every , there holds

(2.11)

3. Proof of Theorem 1.1

Let , and , Bojanic-Cheng decomposition yields

(3.1)

where is defined as in (1.6) and

(3.2)

Obviously, . Thus it follows from (3.1) that

(3.3)

First of all, we estimate

(3.4)

where .

Assuming that , for some (), then we have

(3.5)

Thus

(3.6)

By Lemma 2.3 combining some direct computations, we can easily obtain

(3.7)

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

(3.8)

Thus, by (3.7), (3.8) we have

(3.9)

Finally, we estimate .

First, interval can be decomposed into four parts as

(3.10)

So can be divided into four parts

(3.11)

where .

Noticing and for , we have .

Thus

(3.12)

Next, let .

Now, we recall the Lebesgue-Stieltjes integral representation, and by using partial Lebesgue-Stieltjes integration, we get

(3.13)

An application of (a) in Lemma 2.2 yields

(3.14)

Furthermore, since

(3.15)

we have

(3.16)

Putting in the last integral, we have

(3.17)

It follows from (3.16) and (3.17) that

(3.18)

By a similar method and using Lemma 2.2(b), we obtain

(3.19)

Now, the remainder of our work is to estimate .

For satisfying the growth condition for some positive integer as , we obviously have

(3.20)

Thus, for sufficiently large, there exists a , such that the following inequalities hold:

(3.21)

where . By the definition of the Stirling numbers of the second kind, we readily have

(3.22)

where the Stirling numbers satisfy

(3.23)

Thus we can write

(3.24)

where

(3.25)

From , , we can easily find . For a fixed , when , we have . Thus there holds

(3.26)

Now using the similar method as that in the proof of Lemma  4 of [11], we deduce that

(3.27)

From (3.21), (3.24), and (3.27), we obtain

(3.28)

Finally, by combining (3.12), (3.18), (3.19), and (3.28), we deduce that

(3.29)

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