versión On-line ISSN 1669-9637
Rev. Unión Mat. Argent. v.48 n.1 Bahía Blanca ene./jun. 2007
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.
Later, starting with this integral modification of Bernstein polynomials, Heilmann  first defined modified Lupaş operators (see also Heilmann and Müller  as well as Sinha et al. ). More recently the present author  studied another modification of Lupaş operators. Now we consider Beta operator as a weight function on namely,
denoting the Beta function. Therefore
Let us remark that many years ago W. Meyer-König and K. Zeller  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).
In this section, we shall give certain definition and lemmas which will be used in the sequel.
For every and we have
and there holds the recurrence relation:
where Consequently, for each
Now, using relations
Proof. By using the Leibniz theorem, we obtain
using again Leibniz theorem, we get
On integrating times by parts, we get the required result. □
Proof. Using Taylor's formula, we have
Now, for arbitrary there exists a such that
In order to completely prove the theorem, it is sufficient to show that
Finally we estimate , using the assumption of theorem,
Since is arbitrary, therefore
This completes the proof. □
where the norm is sup-norm over ,
Proof. Applying the Taylor formula
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.
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 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 ]
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.
Recibido: 29 de septiembre de 2006
Aceptado: 28 de marzo de 2007