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### Rev. Unión Mat. Argent. v.48 n.1 Bahía Blanca ene./jun. 2007

Voronovskaya Type Asymptotic Formula For Lupaş-Durrmeyer Operators

Naokant Deo

Abstract. In the present paper, we study some direct results in simultaneous approximation for linear combinations of Lupaş-Beta type operators.

2000 Mathematics Subject Classification. 41A35.

Key words and phrases. Lupaş operator, Linear combinations, Voronovskaya formula.

This research is supported by CAS-TWAS Postdoctoral Fellowship, (Chinese Academy of Sciences, Beijing, China and ICTP, Trieste, Italy).
Permanent address of author: Department of Applied Mathematics, Delhi College of Engineering, Bawana Road, Delhi-110042, India.

1. Introduction

The Bernstein-Durrmeyer (the set of non-negative integers), were introduced by Durrmeyer [2] and independently by Lupaş [5]. For a function they are defined by

where

Later, starting with this integral modification of Bernstein polynomials, Heilmann [3] first defined modified Lupaş operators (see also Heilmann and Müller [4] as well as Sinha et al. [7]). More recently the present author [1] studied another modification of Lupaş operators. Now we consider Beta operator as a weight function on namely,

 (1.1)

where and

denoting the Beta function. Therefore

Let us remark that many years ago W. Meyer-König and K. Zeller [6] have introduced, in order to approximate functions from , the so-called Bernstein-power series defined as

with Because , we see that

The main object of this paper is to establish a Voronovskaya type asymptotic formula and an error estimate for the linear combination of the operators (1.1).

2. Auxiliary Results

In this section, we shall give certain definition and lemmas which will be used in the sequel.

For every and we have

 (2.1)

Lemma 2.1. Let (the set of non-negative integers), we define

then

 (2.2)
 (2.3)

and there holds the recurrence relation:

where Consequently, for each

 (2.5)

Proof. We can easily obtain (2.2) and (2.3) by using the definition of . For the proof of (2.4), we proceed as follows. First

Now, using relations

we obtain

This leads to (2.4). The proof of (2.5) easily follow from (2.2) and (2.4). □

Lemma 2.2. If is differentiable times on then we get

 (2.6)

where

Proof. By using the Leibniz theorem, we obtain

using again Leibniz theorem, we get

Thus

On integrating times by parts, we get the required result. □

3. Voronovskaya Asymptotic Formula

Theorem 3.1. Let integrable in admits its and derivatives, which are bounded at a fixed point and as for some then

where

Proof. Using Taylor's formula, we have

 (3.1)

where

Now, for arbitrary there exists a such that

 (3.2)

Using the value of (2.6) in (3.1), we get

where

In order to completely prove the theorem, it is sufficient to show that

Now

where

and

Using (3.2) and (2.3).

Finally we estimate , using the assumption of theorem,

Thus, from (3.3) and (3.4), we have

Since is arbitrary, therefore

This completes the proof. □

Theorem 3.2. Let and and let be the modulus of continuity of then for

where the norm is sup-norm over ,

and

Proof. Applying the Taylor formula

Thus

Since,

Hence, by Schwartz's inequality

Further, choosing and using Lemma 2.1, we get the required result. □

Acknowledgement. The author is extremely grateful to the referee for making valuable suggestions leading to the better presentation of the paper.

References

[1]    Deo N., Direct result on the Durrmeyer variant of Beta operators, Southeast Asian Bull. Math.,32 (2008) (in press).        [ Links ]

[2]    Durrmeyer J. L., Une formule d'inversion de la transformée de Laplace-applications à la théorie des moments, Thése de 3e cycle, Faculté des Sciences de l' Université de Paris, (1967).        [ Links ]

[3]    Heilmann M., Approximation auf [0,∞) durch das Verfahren der Operatoren vom Baskakov-Durrmeyer Typ, Dissertation, Universität Dortmund, (1987).

[4]    Heilmann M. and Müller M. W., On simultaneous approximation by the method of Baskakov-Durrmeyer operators, Numer. Funct. Anal. and Optimiz., 10(1989), 127-138.        [ Links ]

[5]    Lupaş A., Die Folge der Betaoperatoren, Dissertation, Universität Stuttgart, (1972).        [ Links ]

[6]    Meyer-Köning W. and Zeller K. “Bernsteinsche Poetnzreihen”, Studia Math., 19(1960), 89-94, (see also E. W. Cheney and A. Sharma)

[7]    Sinha R. P., Agrawal P. N. and Gupta V., On simultaneous approximation by modified Baskakov operators, Bull. Soc. Math. Belg. Ser. B, 42(2)(1991), 217-231.        [ Links ]

Naokant Deo
School of Information Science and Engineering, Graduate University of Chinese Academy of Sciences, Zhongguancun Nan Yi Tiao No. 3, Haidian District, Beijing-100080, P. R. China.
dr_naokant_deo@yahoo.com

Recibido: 29 de septiembre de 2006
Aceptado: 28 de marzo de 2007

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