Journal of Calculus of Variations

Volume 2013 (2013), Article ID 814824, 8 pages

http://dx.doi.org/10.1155/2013/814824

Research Article

Some Approximation Properties of -Baskakov-Beta-Stancu Type Operators

Vishnu Narayan Mishra,¹ Kejal Khatri,¹ and Lakshmi Narayan Mishra2,3

¹Department of Applied Mathematics and Humanities, Sardar Vallabhbhai National Institute of Technology, Ichchhanath Mahadev Road, Surat, Gujarat 395 007, India

²L. 1627 Awadhpuri Colony Beniganj, Opp. I.T.I. Ayodhya Main Road, Faizabad, Uttar Pradesh 224 001, India

³Department of Mathematics, National Institute of Technology, Silchar, Cachar District, Assam 788 010, India

Received 7 May 2013; Accepted 8 October 2013

Academic Editor: Adam Kowalewski

Copyright © 2013 Vishnu Narayan Mishra et al. 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.

Abstract.

This paper deals with new type -Baskakov-Beta-Stancu operators defined in the paper. First, we have used the properties of -integral to establish the moments of these operators. We also obtain some approximation properties and asymptotic formulae for these operators. In the end we have also presented better error estimations for the -operators.

1. Introduction

In the recent years, the quantum calculus (-calculus) has attracted a great deal of interest because of its potential applications in mathematics, mechanics, and physics. Due to the applications of -calculus in the area of approximation theory, -generalization of some positive operators has attracted much interest, and a great number of interesting results related to these operators have been obtained (see, for instance, [1–3]). In this direction, several authors have proposed the -analogues of different linear positive operators and studied their approximation behaviors. Also, Aral and Gupta [4] defined -generalization of the Baskakov operators and investigated some approximation properties of these operators. Subsequently, Finta and Gupta [5] obtained global direct error estimates for these operators using the second-order Ditzian Totik modulus of smoothness. To approximate Lebesgue integrable functions on the interval , modified Beta operators [6] are defined as

where and .

The discrete -Beta operators are defined as

Recently, Maheshwari and Sharma [7] introduced the -analogue of the Baskakov-Beta-Stancu operators and studied the rate of approximation and weighted approximation of these operators. Motivated by the Stancu type generalization of -Baskakov operators, we propose the -analogue of the operators , recently introduced and studied for special values by Gupta and Kim [8] as

where and .

We know that and . We mention that (see [8]).

Very recently, Gupta et al. [9] introduced some direct results in simultaneous approximation for Baskakov-Durrmeyer-Stancu operators. The aim of this paper is to study the approximation properties of a new generalization of the Baskakov type Beta Stancu operators based on -integers. We estimate moments for these operators. Also, we study asymptotic formula for these operators. Finally, we give better error estimations for the operator (3). First, we recall some definitions and notations of -calculus. Such notations can be found in [10, 11]. We consider as a real number satisfying . For ,

The -binomial coefficients are given by

The -derivative of a function is given by

The -analogues of product and quotient rules are defined as

The -Jackson integrals and the -improper integrals are defined as [12, 13]

provided that the sums converge absolutely. Using (9), De Sole and Kac [14] defined the -analogue of Beta functions of second kind as follows:

where . This function is -constant in ; that is, . It was observed in [14] that is independent of ; this is because from the integral and the term cancels out. In particular for any positive integer , we have

Also, we have

In [8], Gupta and Kim obtained recurrence formula for the moments of the operators as follows.

Theorem 1 (see [8]). If one defined the central moments as

then, for , one has the following recurrence relation:

2. Moment Estimates

Lemma 2 (see [8]). The following equalities hold.

(i).(ii).(iii) , for .

Lemma 3. The following equalities hold.

(i).(ii), for .(iii) , for .

Proof. The operators are well defined on function , , . By Lemma 2, for every and , we have

Similarly,

Remark 4. For all , , we have the following recursive relation for the images of the monomials under in terms of , , as

Remark 5. If we put and , we get the moments of the modified Beta operators [6] as

Remark 6. From Lemma 3, we have

3. Direct Result and Asymptotic Formula

Let the space of all real-valued continuous bounded functions be endowed with the norm . Further, let us consider the following -functional:

where and . By [15, page 177, Theorem  2.4], there exists an absolute constant such that

where

is the second-order modulus of smoothness of . Also we set

Theorem 7. Let and such that as . Then for all and , there exists an absolute constant such that

Proof. We are introducing the auxiliary operators as follows:

From (25) and Lemma 3, we have

Let and . Using Taylor’s formula

applying , and by (26), we get

On the other hand, since

We conclude by Remark 6 that

From (25), we can write that

Now, taking into account the boundedness of and from (31), we get

Now, taking infimum on the right-hand side over all and from (21), we get

where . This proves the theorem.

Our next result in this section is an asymptotic formula.

Theorem 8. Let be bounded and integrable function on the interval ; the second derivative of exists at a fixed point and such that as . Consider

Proof. Using Taylor’s expansion of , we can write

where is bounded and . Applying the operator to the above relation, we get

where and are defined in Remark 6.

Using Cauchy-Schwarz inequality, we have

Using Theorem 1 with the help of Remark 4, we can easily find that

Also, since

Thus,

which completes the proof.

4. Better Estimation

It is well known that the operators preserve constant as well as linear functions. To make the convergence faster, King [16] proposed an approach to modify the classical Bernstein polynomials, so that this sequence preserves two test functions: and . After this, several researchers have studied that many approximating operators, , possess these properties; that is, , where    or   , for example, Bernstein, Baskakov, and Baskakov-Durrmeyer-Stancu operators (see [4, 5, 17–19]).

As the operators introduced in (3) preserve only the constant functions, further modification of these operators is proposed to be made so that the modified operators preserve the constant as well as linear functions. For this purpose, the modification of is as follows:

where

Lemma 9. For each , one has

Lemma 10. For each , the following equalities hold:

Theorem 11. Let and such that as . Then for all and , there exists an absolute constant such that

Proof. Let and . Using Taylor’s formula, we get

Applying , we get

Obviously, we have

Therefore,

Since , thus

Now, taking infimum on the right-hand side over all and from (21), we get

which proves the theorem.

Theorem 12. Let be bounded and integrable function on the interval ; the second derivative of exists at a fixed point and such that as ; then

The proof follows along the same lines of Theorem 8.

Conflict of Interests

The authors declare that there is no conflict of interests.

Acknowledgments

The authors thank the anonymous learned reviewers for their valuable suggestions, which substantially improved the standard of the paper. Special thanks are due to Professor Adam Kowalewski, for kind cooperation and smooth behavior during communication and for his efforts to send the reports of the paper timely.

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