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## Revista de la Unión Matemática Argentina

*versión impresa* ISSN 0041-6932

### Rev. Unión Mat. Argent. vol.50 no.1 Bahía Blanca jun. 2009

**Approximation degree for generalized integral operators**

**S. Jain and R. K. Gangwar**

* Abstract.* Very recently Jain et al. [4] 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 [3] is defined as

| (1) |

where

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 [4] 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 [3]. For certain operators Bojanic and Khan [2] and Taberska [10] 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 [1],[5], [6] and [8]. Very recently Ispir et al. [9] 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 [4].** *Let the* *th order central moment be of the operators (1)* *be defined by*

*Then, we have*

*and*

*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

| (2) |

**Remark 2.** In view of Remark 1, it can be observed by Holder's inequality that

| (3) |

**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

Also

and

Thus we can write

| (4) |

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

Thus

| (5) |

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):

| (7) |

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).

We define

**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 [7], 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.

[1] R. Bojanic and F. Cheng, Rate of convergence of Bernstein polynomials for functions with derivatives of bounded variation, J. Math. Anal. Appl. 141 (1989), no. 1, 136-151. [ Links ]

[2] R. Bojanic and M. K. Khan, Rate of convergence of some operators of functions with derivatives of bounded variation, Atti. Sem. Mat. Fis. Univ. Modena (2)39 (1991), 495-512. [ Links ]

[3] R. K. Gangwar and V. K. Jain, Rate of approximation for certain generalized operators, General Math., to appear. [ Links ]

[4] S. Jain, R. K. Gangwar and D. K. Dubey, Convergence for certain Baskakov Durrmeyer type operators, Nonlinear Functional Anal. Appl.,to appear. [ Links ]

[5] V. Gupta, U. Abel and M. Ivan, Rate of convergence of Beta operators of second kind for functions with derivatives of bounded variation, Int. J. Math. Math. Sci. 2005(23) (2005), 3827-3833. [ Links ]

[6] V. Gupta and P. N. Agrawal, Rate of convergence for certain Baskakov Durrmeyer type operators, Anal. Univ. Ordea Fasc. Math. 14 (2007), 33-39. [ Links ]

[7] V. Gupta and H. Karsli, Rate of convergence for the Bezier variant of the MKZD operators, Georgian Math. J. 14 (2007), 651-659. [ Links ]

[8] V. Gupta, V. Vasishtha and M. K. Gupta, Rate of convergence of summation-integral type operators with derivatives of bounded variation, JIPAM. J. Inequal. Pure Appl. Math. 4(2) (2003), Art.34. [ Links ]

[9] N. Ispir, A. Aral and O. Dogru, On Kantorovich process of a sequence of the generalized linear positive operators, Numer. Funct. Anal. Optim., 29(5-6) (2008), 574-589. [ Links ]

[10] P. Pych Taberska, Pointwise approximation of absolutely continuous functions by certain linear operators, Funct. Approx. Comment. Math. 25 (1997), 67-76. [ Links ]

**S. Jain**** ** Guru Nanak Institute of Management

Road No. 75, Punjabi Bagh, New Delhi 110026, India

jainshipra11@rediffmail.com

**Ravindra Kumar Gangwar**** ** Department of Mathematics, Bareilly College

Bareilly 243001, India

ravindra1402@yahoo.co.in

**Recibido**: 6 de marzo de 2008

**Aceptado**: 7 de octubre de 2008