Revista de la Unión Matemática Argentina
versión ISSN 0041-6932
Rev. Unión Mat. Argent. vol.50 no.1 Bahía Blanca jun. 2009
S. Jain and R. K. Gangwar
Abstract. Very recently Jain et al.  proposed generalized integrated Baskakov operators and estimated some approximation properties in simultaneous approximation. In the present paper we establish the rate of convergence of these operators and its Bezier variant, for functions which have derivatives of bounded variation.
For and the general family of Baskakov type operators considered in  is defined as
In the alternative form the above operators (1), can be defined as
where the kernel in terms of Dirac delta function is given by
We define then as a special case we have Let be the class of absolutely continuous functions defined on satisfying the growth condition and having a derivative on the interval coinciding a.e. with a function which is of bounded variation on every finite subinterval of It can be observed that all functions posses for each a representation
In  the authors studied some direct results in simultaneous approximation for the operators (1). Very recently the rate of convergence for bounded function for the operators has been obtained by Gangwar and Jain . For certain operators Bojanic and Khan  and Taberska  estimated the rate of convergence for functions having derivative of B.V. The analogous problem on the convergence rate for the Bernstein polynomials and certain other integral operators were studied in ,,  and . Very recently Ispir et al.  considered the Kantorovich process of a generalized sequence of linear positive operators and estimated the rate of convergence for absolutely continuous functions having a derivative coinciding a.e., with a function of bounded variation.
As the operators defined by (1) are the generalized operators, this motivated us to extend the studies and here we study the approximation properties of the operators and its Bezier variant. We estimate the convergence rate for functions whose first derivative is of bounded variation.
We shall use the following lemmas to prove our main theorem.
Lemma 1 . Let the th order central moment be of the operators (1) be defined by
Then, we have
and for we have
It is easily checked that for all one has
Remark 1. In particular given any number and , by Lemma 1, we have for n sufficiently large one has
Remark 2. In view of Remark 1, it can be observed by Holder's inequality that
Lemma 2. Let and the kernel is defined by (1), then for sufficiently large, we have
Proof. First we prove (i), by (2), we have
The proof of (ii) is similar, we omit the details.
In this section, we prove the following main theorem.
Theorem 1. Let , and . Then for and for sufficiently large, we have
where denotes the total variation of on and is defined by
Proof. Using the fact that we can write
Also, we can write
Next, we have thus
Thus we can write
To complete the proof of the theorem it is sufficient to estimate the terms and . Applying integration by parts, using Lemma 2 and taking ,we have
Let . Then we have
On the other hand, we have
Next applying Holder's inequality, and Lemma 1, we proceed as follows for the estimation of the first two terms in the right hand side of (6):
Also the third term of the right side of (6) is estimated as
Collecting the estimates (4)-(7), we get the required result.
This completes the proof of Theorem 1.
For or , the Bezier variant of the operators (1) can be defined as
where and the kernel is given by
being Dirac delta function. In case the operators defined by (8) reduce to the operators (1).
Remark 3. By Lemma 2 for we can write
Remark 4. In view of Lemma 1 and Remark 1, for it can be observed that
Our main result is stated as follows:
Theorem 2.Let , also suppose and . Then for and sufficiently large, we have
where denotes the total variation of on and the auxiliary function is as given in Theorem 1.
Proof. Following the methods presented in , we can write
Using Remark 4, we have
In order to complete the proof of the theorem, it is sufficient to estimate the terms and given in (9) above. Using Remark 3 and proceeding along the lines of proof of Theorem 1, we get the desired estimate. Here we omit the details.
We are thankful to the referee for his critical review leading to overall improvement of the paper.
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Guru Nanak Institute of Management
Road No. 75, Punjabi Bagh, New Delhi 110026, India
Ravindra Kumar Gangwar
Department of Mathematics, Bareilly College
Bareilly 243001, India
Recibido: 6 de marzo de 2008
Aceptado: 7 de octubre de 2008