Abstract and Applied Analysis

Volume 2013 (2013), Article ID 732069, 8 pages

http://dx.doi.org/10.1155/2013/732069

Research Article

Degree of Approximation by Hybrid Operators

Naokant Deo,¹ Hee Sun Jung,² and Ryozi Sakai³

¹Department of Applied Mathematics, Delhi Technological University (Formerly Delhi College of Engineering), Bawana Road, Delhi 110042, India

²Department of Mathematics Education, Sungkyunkwan University, Seoul 110-745, Republic of Korea

³Department of Mathematics, Meijo University, Nagoya 468-8502, Japan

Received 8 May 2013; Accepted 12 July 2013

Academic Editor: Irena Rachůnková

Copyright © 2013 Naokant Deo 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.

We consider hybrid (Szász-beta) operators, which are a general sequence of integral type operators including beta function, and we give the degree of approximation by these Szász-beta-Durrmeyer operators.

1. Introduction

The Lupaş-Durrmeyer operators were introduced by Sahai and Prasad [1] who studied the asymptotic formula for simultaneous approximation, and many mathematicians have given different results for the Durrmeyer operators (see [2–6]). Now we consider here a sequence of linear positive operators, which was introduced by Gupta et al. [7] as follows. Let and be positive integers. For satisfying ,

where is a positive integer,

Let . For a function on , we define the norm by

Recently Jung and Sakai [8] investigated the Lupaş-Durrmeyer operators and studied the circumstances of convergence. Motivated with the idea of Jung and Sakai [8], we give the degree of approximation by Szász-Beta-Durrmeyer operators in this paper.

2. Basic Results

Lemma 1 (cf. [7]). Let , , , and be integers with , , and :

Then one has

(i) and ,(ii)for

(iii)

where is a polynomial of degree such that the coefficients of are bounded independently of .

Proof. Let . Then (i)

Using

we see that

(ii) Using , we obtain

Since we know that

we have

Then substituting (12) into (10), we consider the following:

Then since we see

we have

Here the last equation follows from integration by parts. Furthermore, we easily see

Therefore, we conclude

Consequently, (ii) is proved.

(iii) For , (6) holds. Let us assume (6) for . We note

So, we have, by the assumption of induction,

Here, if is even, then

and if is odd, then

Hence we have

and here we see that is a polynomial of degree such that the coefficients of are bounded independently of .

Lemma 2 (cf. [7]). Let , , and be integers with . Let satisfy for a positive integer

Then one has, for ,

where

Proof. Using

we have

3. Main Results

Theorem 3. Let , and let and be nonnegative integers. Let and be integers with . Let satisfy

Then one has uniformly, for and ,

Proof. Let and . By the second inequality of (28),

Let . Then using Lemma 2 and

we have

From (30) and Lemma 1, we have

Next, we estimate . By the use of the first inequality in (28), we have

Now using and the notation

we have

Then, with ,

Here for , we get

because

Finally we get

From (32),

If we put , then we get

In the following, we let ,.

Theorem 4. Let and be nonnegative integers. Let and be integers with . Let satisfy

Then one has uniformly, for and ,

Proof. For , we have

From (45), (46), and Lemma 2, we get

Using , we obtain

Therefore, we have

For , we have , , and . Hence

Let us define the weighted modulus of smoothness by

where

Theorem 5. Let and be nonnegative integers. Let and be integers with . Then one has, for ,

To prove Theorem 5, we need the following theorem.

Theorem 6. Let and be nonnegative integers. Let and be integers with . Let satisfy

Then one has uniformly, for , , and ,

Proof. Using , we have

Therefore, by Lemma 1 (6), we have

Since is uniformly bounded on , we have with Lemma 2 and (59)

Therefore, we have the result.

The Steklov function for is defined as follows:

Then for the Steklov function with respect to , we have the following properties.

Lemma 7 (see [8, Lemma 2.4]). Let , and let be a positive and nonincreasing function on . Then

(i),(ii)

(iii)

(iv)

Now, we prove Theorem 5.

Proof of Theorem 5. We know that, for ,

Then first, we split it as follows:

Then for the first term, we have, using Theorem 6, (62), and (65),

Here, we suppose , and then we know that

For the second term, from Theorem 4, (65), (63), and (64) of Lemma 7,

Therefore, we have

If we let , then

because .

Acknowledgment

The authors thank the referees for many kind suggestions and comments.

References

A. Sahai and G. Prasad, “On simultaneous approximation by modified Lupas operators,” Journal of Approximation Theory, vol. 45, no. 2, pp. 122–128, 1985.

N. Deo, “Faster rate of convergence on Srivastava-Gupta operators,” Applied Mathematics and Computation, vol. 218, no. 21, pp. 10486–10491, 2012.

N. Deo and N. Bhardwaj, “Some approximation results for Durrmeyer operators,” Applied Mathematics and Computation, vol. 217, no. 12, pp. 5531–5536, 2011.

M. M. Derriennic, “Sur l'approximation de fonctions intégrables sur [0, 1] par des polynômes de Bernstein modifiés,” Journal of Approximation Theory, vol. 31, no. 4, pp. 325–343, 1981.

O. Duman, M. A. Özarslan, and H. Aktuğlu, “Better error estimation for Szász-Mirakjan-beta operators,” Journal of Computational Analysis and Applications, vol. 10, no. 1, pp. 53–59, 2008.

H. S. Kasana, G. Prasad, P. N. Agrawal, and A. Sahai, “Modified Szász operators,” in Mathematical Analysis and Its Applications (Kuwait, 1985), vol. 3 of KFAS Proceedings Series, pp. 29–41, Pergamon, Oxford, UK, 1988.

V. Gupta, G. S. Srivastava, and A. Sahai, “On simultaneous approximation by Szász-beta operators,” Soochow Journal of Mathematics, vol. 21, no. 1, pp. 1–11, 1995.

H. S. Jung and R. Sakai, “Approximation by Lupas-type operators and Szász-Mirakyan-type operators,” Journal of Applied Mathematics, vol. 2012, Article ID 546784, 28 pages, 2012.