ISRN Mathematical Analysis

VolumeΒ 2011, Article IDΒ 184374, 11 pages

http://dx.doi.org/10.5402/2011/184374

## On -Approximation by Iterative Combination of Bernstein-Durrmeyer Polynomials

^{1}Department of Mathematics, SMD College Poonpoon, Patna, Bihar, India^{2}Department of Mathematics, IIT Roorkee, Roorkee 247667, India

Received 11 November 2010; Accepted 9 December 2010

Academic Editors: O.Β GuΓ¨s and R.Β Stenberg

Copyright Β© 2011 T. A. K. Sinha et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We improve the degree of approximation by Bernstein-Durrmeyer polynomials taking their iterates and obtain error estimate in higher-order approximation.

#### 1. Introduction

The Bernstein-Durrmeyer polynomials where , , were introduced by Durrmeyer [1] and extensively studied by Derriennic [2] and several other researchers. It turns out that the order of approximation by these operators is, at best, however smooth the function may be. In order to improve this rate of approximation, we consider an iterative combination of the operators . This technique of improving the rate of convergence was given by Micchelli [3] who first used it to improve the order of approximation by Bernstein polynomials . Recently, this technique has been applied to obtain some direct and inverse theorems in ordinary and simultaneous approximation by several sequences of linear positive operators in uniform norm (c.f., e.g., [4β6]). The object of this paper is to study some direct theorems in -approximation by the operators .

For , the operators can be expressed as where is the kernel of the operators.

For (the set of nonnegative integers), the th order moment for the operators is defined as

Let , , , and denote the classes of bounded Lebesgue integrable, infinitely differentiable, absolutely continuous functions, and functions of bounded variations, respectively, on the interval .

The Iterative combination of the operators is defined as where and for .

In Section 2 of this paper, we give some definitions and auxiliary results which will be needed to prove the main results. In Section 3, we obtain an estimate of error in -approximation by the iterative combination in terms of -norm of derivatives of the function. From these estimates, we obtain a general error estimate in terms of th integral modulus of smoothness of the function.

In what follows, we suppose that and , . Further, is a constant not always the same.

#### 2. Preliminaries and Auxiliary Results

In the sequel, we will require the following results.

Lemma 2.1 (see [5]). *For the function , one has , , and there holds the recurrence relation
**
for . **Consequently, we have *(i)* is a polynomial in of degree , *(ii)*for every , , where is the integer part of .*

The th order moment for the operator is defined as , (the set of natural numbers). We denote by .

Lemma 2.2 (see [1]). *For the function , there holds the result
**
where are certain polynomials in independent of and .*

Lemma 2.3 (see [7]). *There holds the recurrence relation
*

Lemma 2.4 (see [7]). *For , , there holds .*

Using Lemmas 2.1 and 2.3, we can prove the following.

Lemma 2.5. *For , , and , we have
*

Let , , and . Then, for sufficiently small the Steklov mean of th order corresponding to is defined as follows: where is the forward difference operator with step length .

Lemma 2.6. *Let , , and . Then, for the function , we have *(a)* has derivatives up to order over , *(b)*, ,*(c)*, *(d)*,
*(e)*,
**where is a constant that depends on but is independent of and .*

Following [8, Theoremββ18.17] or [9, pages 163β165], the proof of the above lemma easily follows hence the details are omitted.

Let , . Then, the Hardy-Littlewood majorant of the function is defined as

Lemma 2.7. *If , , then and
*

The lemma follows from [10, page 32].

The next lemma gives a bound for the intermediate derivatives of in terms of the highest-order derivative and the function in -norm.

Lemma 2.8 (see [11]). *Let , . Suppose and . Then,
**
where are certain constants independent of .*

The dual operator corresponding to the operator is defined as Then, the corresponding th order moment is given by .

Lemma 2.9. *For the function , there holds the recurrence relation
*

*Proof. *In view of the relation , we get
Expanding as a polynomial in and integrating by parts, we get

Rearrangement of the terms gives (2.10).

*Remark 2.10. *From (2.10), it follows that , where is the integer part of .

#### 3. Main Result

In this section, we obtain an error estimate in terms of norm. The proof of the case makes use of Lemma 2.7 regarding Hardy-Littlewood majorant and Lemma 2.8, while for , we require only Lemma 2.8.

Theorem 3.1. *If , , has derivatives of order on with , and , then for sufficiently large **
Moreover, if , has derivatives up to the order on with , and , then for sufficiently large there holds
**
where is a certain constant independent of and .*

*Proof. *Let , then for all and , we can write
where is the characteristic function of the interval and
Therefore, operating by on both sides of (3.3), we obtain three terms, say , , and corresponding to the three terms in the right-hand side of (3.3).

In view of Lemmas 2.4 and 2.8, we get
Let be the Hardy-Littleood majorant of on . Then, in order to estimate , it is sufficient to consider the estimate for
Applying HΓΆlder's inequality, Lemma 2.1, and then Fubini's theorem, we get
Now, in view of Lemmas 2.1 and 2.7, we have
Consequently,
For , , we can find a such that . Thus,
On an application of HΓΆlder's inequality, Lemma 2.1, and Fubini's theorem, we get
Now in view of Lemmas 2.1 and 2.8, we have the inequality
Combining the estimates (3.5)β(3.12), (3.1) follows.

Now, let . Then, we can expand for almost all and for all , as
where and are defined as above. Therefore, operating by on both sides of (3.13), we obtain three terms , , and , say corresponding to the three terms in the right-hand side of (3.13).

Now proceeding as in the case of the estimate of , we have

It can easily be shown that
Consequently, by induction, we get
Therefore, in order to get an estimate for , it is sufficient to consider the estimate for
For each , there exists the integer s.t.

To estimate , for all , , we can choose a such that . Therefore, we get the inequality
Since, is symmetric in and , there follows
In view of Lemma 2.8, we obtain
From the estimates (3.14)β(3.21) and the definition of , we get (3.2).

Theorem 3.2. *If , . Then, for all sufficiently large there holds
**
where is a constant independent of and .*

*Proof. *In order to prove the theorem, it is sufficient to prove it for the function , where be such that and in . Let for convenience . Let be the Steklov mean of order corresponding to the function , where is sufficiently small. Then, we have
Let be the characteristic function of . Then,
For using HΓΆlder's inequality and then applying Fubini's theorem, we get
This in view of property (c) of Steklovβs mean implies
In the case , (3.22) is obtained by boundedness of the operator . Now, as above in Theorem 3.1 for , we can choose a such that . Then, from Fubini's theorem and moment estimates 1 of dual operator, we get
Therefore,
Consequently, we have the estimate
Now using Theorem 3.1 and Lemma 2.8,
In view of property (c) of Steklovβs means, we get the inequality
Choosing , the result follows from the estimates of .

#### Acknowledgment

A. R. Gairola is thankful to the *Council of Scientific and Industrial Research*, New Delhi, India for financial support to carry out the above work.

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