Journal of Calculus of Variations

Volume 2013 (2013), Article ID 489249, 8 pages

http://dx.doi.org/10.1155/2013/489249

Research Article

Some Approximation Properties of Modified Jain-Beta Operators

Vishnu Narayan Mishra1,2 and Prashantkumar Patel1,3

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

²L. 1627 Awadh Puri Colony Beniganj, Phase-III, Opp. I.T.I., Ayodhya Main Road Faizabad, Uttar Pradesh 224 001, India

³Department of Mathematics, St. Xavier College, Ahmedabad, Gujarat 380 009, India

Received 21 May 2013; Revised 24 October 2013; Accepted 3 December 2013

Academic Editor: Jacob Engwerda

Copyright © 2013 Vishnu Narayan Mishra and Prashantkumar Patel. 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.

Generalization of Szász-Mirakyan operators has been considered by Jain, 1972. Using these generalized operators, we introduce new sequences of positive linear operators which are the integral modification of the Jain operators having weight functions of some Beta basis function. Approximation properties, the rate of convergence, weighted approximation theorem, and better approximation are investigated for these new operators. At the end, we generalize Jain-Beta operator with three parameters , , and and discuss Voronovskaja asymptotic formula.

1. Introduction

For ,  , let

then

Equation (1) is a Poisson-type distribution which has been considered by Consul and Jain [1].

In 1970, Jain [2] introduced and studied the following class of positive linear operators:

where and has been defined in (1).

The parameter may depend on the natural number . It is easy to see that ; (3) reduces to the well-known Szász-Mirakyan operators [3]. Different generalization of Szász-Mirakyan operator and its approximation properties is studied in [4, 5]. Kantorovich-type extension of was given in [6]. Integral version of Jain operators using Beta basis function is introduced by Tarabie [7], which is as follows:

In Gupta et al. [8] they considered integral modification of the Szász-Mirakyan operators by considering the weight function of Beta basis functions. Recently, Dubey and Jain [9] considered a parameter in the definition of [8]. Motivated by such types of operators we introduce new sequence of linear operators as follows:

For and ,

where is defined in (1) and

The operators defined by (5) are the integral modification of the Jain operators having weight function of some Beta basis function. As special case, the operators (5) reduced to the operators recently studied in [7]. Also, if and , then the operators (5) turn into the operators studied in [8].

In the present paper, we introduce the operators (5) and estimate moments for these operators. Also, we study local approximation theorem, rate of convergence, weighted approximation theorem, and better approximation for the operators . At the end, we propose Stancu-type generalization of (5) and discuss some local approximation properties and asymptotic formula for Stancu-type generalization of Jain-Beta operators.

2. Basic Results

Lemma 1 (see [2]). For ,  , one has

Lemma 2. The operators , defined by (5) satisfy the following relations:

Proof. By simple computation, we get

Lemma 3. For , , and with , one has

(i),(ii) .

Lemma 4. For , , one has

Proof. Since , , and , we have

which is required.

3. Some Local Approximation

Let ,   being a constant depending on . By , we denote the subspace of all continuous functions belonging to . Also, is subspace of all the function for which is finite. The norm on is .

If we look at Lemma 2 and based on the famous Korovkin theorem [10], it is clear that does not form an approximation process. To do this approximation process, we replace constant by a number . If

then Lemma 2 gives

uniformly on any compact interval . Based on Korovkin’s criteria we state the following.

Theorem 5. Let with be defined in (5), where . For any compact and for each one has

Now, we establish a direct local approximation theorem for the modified operators in ordinary approximation. Let the space of all continuous and bounded functions be endowed with the norm . Further let us consider the following -functional:

where and . By the methods given in [11], there exists an absolute constant such that

where

is the second order modulus of smoothness of .

Theorem 6. For and , one has

where is a positive constant.

Proof. We introduce the auxiliary operators as follows:

Let and . By Taylor’s expansion we have

Applying , we get

Applying Lemma 2, we get

Since

Taking infimum overall , we get

In view of (18)

which proves the theorem.

4. Rate of Convergence and Weighted Approximation

For any positive , by

we denote the usual modulus of continuity of on the closed interval . We know that, for a function , the modulus of the continuity tends to zero.

Now we give a rate of convergence theorem for the operator .

Theorem 7. Let and be its modulus of continuity on the finite interval , where . Then, for ,

Proof. For and , since , we have

For and , we have

with .

From (31) and (32) we can write

for and . Thus,

Hence, by Schwarz’s inequality and Lemma 3, for ,

By taking , we get

which proves the theorem.

Now we will discuss the weighted approximation theorem, where the approximation formula holds true on the interval .

Theorem 8. If , , and , then,

Proof. Using the theorem in [12] we see that it is sufficient to verify the following three conditions:

Since , the first condition of (38) is fulfilled for . By Lemma 2 we have

and the second condition of (38) holds for as with .

Similarly, we can write, for ,

which implies that

Thus, the proof is completed.

5. Better Error Approximation

In this section, we modified operator (5), in such way that the linear functions are preserved. The technique, which replaced by appropriate function, was studied for many operators, for example, Bernstein, Szász, Szász-Beta operators, and so on [13–20].

We start by defining

We note that , for any . By replacing by we give the following modification of the operators :

where

and , ; the term is given in (5).

Lemma 9. For and , one has

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

Lemma 10. For , , and with , one has

(i),(ii).

Lemma 11. For , , one has

Proof. Since and , we have

which is required.

Theorem 12. Let . Then for and , one has

Proof. Let and . Using Taylor’s expansion

and Lemma 10, we have

Also, . Thus,

Since ,

Finally taking the infimum on right side over all , we get

In view of (18), we obtain

which proves the theorem.

Remark 13. We claim that the error estimation in Theorem 12 is better than that of (20), provided and . Indeed, in order to get this better estimation we must show that . One can obtain that

Also,

which holds true, with and . Thus, .

6. Stancu Approach

In 1968, Stancu introduced Bernstein-Stancu operator, which is a linear positive operator with two parameters and satisfying the condition . Inspired by the Stancu-type generalization of Bernstein operator and the recent important work on several other operators are discuss in [21–27], we propose following modification of the operator as

where and are defined in (5).

Lemma 14. For ,  , the following inequalities holds:

(i), (ii), (iii)   +       + .

The proof of the above lemma can be obtained by using linearity of operators and Lemma 2.

Lemma 15. If one denotes central moments by ,  , then one has

Theorem 16. Let with be defined in (56), where . For any compact and for each , one has

The proof is based on Korovkin’s criterion and Lemma 14.

Theorem 17. Let and , one has

for every and , where is a positive constant.

The proof of Theorem 17 is just similar to Theorem 6.

Now, we establish the Voronovskaja-type asymptotic formula for the operators .

Theorem 18. Let be bounded and integrable on , first and second derivatives of exist at a fixed point , and such that as ; then

Proof. Let , , and be fixed. By Taylor’s expansion we can write

where is Peano form of the remainder, , and .

Applying to the previous, we obtain

By Cauchy-Schwarz’s inequality, we have

Observe that and . Then it follows that

uniformly with respect to .

Now, from (63) and (64), we obtain .

Using as , we obtain

Using above limits, we have

which proves the theorem.

Remark 19. In particular, if and , then the operators , such that as , reduce to the Jain-Beta operators recently introduced by Tarabie [7]. We obtain the following conclusion of the above asymptotic formula for the Jain-Beta operator in the ordinary approximation as follows:

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to express their deep gratitude to the anonymous learned referees for their valuable suggestions and comments. The second author is thankful to the Department of Mathematics, St. Xavier College, Ahmedabad, Gujarat, for carrying out his research work under the supervision of Dr. Vishnu Narayan Mishra at SVNIT, Surat, Gujarat, India. Special thanks are due to Professor Jacob C. Engwerda for kind cooperation and smooth behavior during communication and for the efforts to send the reports of the paper timely.

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