On Rate of Approximation by Modified Beta Operators

Abstract

We establish the rate of convergence for the modified Beta operators , for functions having derivatives of bounded variation.

1. Introduction

In the year 1995 Gupta and Ahmad [1] proposed modified Beta operators so as to approximate Lebesgue integrable functions on as where These operators are linear positive operators and reproduce only the constant functions. We set In particular, it can be observed by the definition of Beta function that

Gupta and Ahmad [1] estimated asymptotic formula and error estimate for these operators in simultaneous approximation. Later Gupta et al. [2] studied the approximation properties of these operators in norm and they obtained some direct results for the linear combinations. Here we continue our studies on these operators. First we consider the following class of functions.

By we mean the class of absolutely continuous functions defined on , which satisfy the following:

Very recently Gupta and Agrawal [3] and Gupta et al. [4] have obtained an interesting result on the rate of convergence of certain Durrmeyer type operators in ordinary and simultaneous approximation. spir et al. [5] estimated similar results for Kantorovich operators for functions with derivatives of bounded variation. We now extend the study for the modified Beta operators and estimate the rate of convergence of the operators (1) for functions having derivatives of bounded variation.

2. Auxiliary Results

We shall use the following lemmas to prove our main theorem.

Lemma 2.1 ([1]). Let the function , , be defined as Then Also for each , one has

Remark 2.2. From Lemma 2.1 and using Cauchy-Schwarz inequality, for sufficiently large, it follows that where .

Lemma 2.3. Let , then for and sufficiently large, one has

Proof. By using Lemma 2.1, for sufficiently large, we have The proof of the second inequality is similar, and we skip the details.

3. Rate of Approximation

Our main result is stated as follows.

Theorem 3.1. Let ,??, and . Then for sufficiently large, one has where the auxiliary function is given by denotes the total variation of on

Proof. By mean value theorem, we have Applying the identity Equation (3.3) becomes (keeping in mind that the last term of the above identity vanishes) As Thus in view of the above values, Lemma 2.1, and Remark 2.2, (3.5) reduces to In order to complete the proof of the theorem, it is sufficient to estimate the terms and Applying integration by parts, using Lemma 2.3 and taking , we have Let Then we have Thus, On the other hand, we have For the estimation of the first two terms in the right-hand side of (3.11).we proceed as follows.
Applying Holder's inequality, Remark 2.2, and Lemma 2.1, we have Also, by Remark 2.2, the third term of the right side of (3.11) is given by Thus Combining the estimates (3.7), (3.10), and (3.14) we get the desired result. This completes the proof of the theorem.

References

1. V. Gupta and A. Ahmad, “Simultaneous approximation by modified beta operators,” İstanbul Üniversitesi Fen Fakültesi Matematik Dergisi, vol. 54, pp. 11–22, 1995. View at: Google Scholar | MathSciNet
2. V. Gupta, P. Maheshwari, and V. K. Jain, “$L_p$-inverse theorem for modified beta operators,” International Journal of Mathematics and Mathematical Sciences, vol. 2003, no. 20, pp. 1295–1303, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
3. V. Gupta and P. N. Agrawal, “Rate of convergence for certain Baskakov Durrmeyer type operators,” Annals of Oradea University, Fascicola Matematica, vol. 14, pp. 33–39, 2007. View at: Google Scholar | Zentralblatt MATH | MathSciNet
4. V. Gupta, P. N. Agrawal, and A. R. Gairola, “On the integrated Baskakov type operators,” Applied Mathematics and Computation, vol. 213, no. 2, pp. 419–425, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
5. N. İspir, A. Aral, and O. Doğru, “On Kantorovich process of a sequence of the generalized linear positive operators,” Numerical Functional Analysis and Optimization, vol. 29, no. 5-6, pp. 574–589, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

Copyright

Copyright © 2009 Prerna Maheshwari (Sharma) and Vijay Gupta. 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.