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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 614810, 16 pages
http://dx.doi.org/10.1155/2012/614810
Research Article

Approximation of the Summation-Integral-Type -Szász-Mirakjan Operators

1Department of Mathematics and Computer Science, Wuyi University, Wuyishan 354300, China
2Department of Mathematics, Xiamen University, Xiamen 361005, China

Received 26 July 2012; Revised 20 September 2012; Accepted 24 October 2012

Academic Editor: Behnam Hashemi

Copyright © 2012 Mei-Ying Ren and Xiao-Ming Zeng. 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 introduce summation-integral-type -Szász-Mirakjan operators and study approximation properties of these operators. We establish local approximation theorem. We give weighted approximation theorem. Also we estimate the rate of convergence of these operators for functions of polynomial growth on the interval .

1. Introduction

First of all, we mention some concepts and notations from -calculus. All of the results can be found in [15]. In what follows, is a real number satisfying .

For each nonnegative integer , the -integer and the -factorial are defined as

Then for integers , , we have

For the integers , the -binomial coefficient is defined as

The two -analogues of the exponential function are defined as where .

It is easily observed that

The -Jackson integral and the -improper integral are defined as provided the sums converge absolutely.

For , the -Gamma function is given by where

It was observed in [5] that is a -constant, that is, . In particular, for , and . , .

In 1997, Phillips [6] firstly introduced and studied analogue of Bernstein polynomials. After this, the applications of -calculus in the approximation theory become one of the main areas of research; many authors studied new classes of -generalized operators (for instance, see [712]). In order to approximate integrable functions, in 2008 Gupta and Heping [13] introduced and studied the -Durrmeyer-type operators as follows: where , , , , and

In 2010, Mahmudov and Kaffaoğlu [14] defined and studied -Szász-Durrmeyer operators as follows: where , , , and

In 2011, Aral and Gupta [15] introduced and studied -generalized Szász-Durrmeyer operators by means of the -integral as follows: where , , , is a sequence of positive numbers such that , , and

In the present paper, we will introduce a kind of -analogue of the summation-integral-type Szász-Mirakjan operators as follows.

Let . For every , , , and , the summation-integral-type -Szász Mirakjan operators are defined as where

We will study the approximation properties of the operators (1.15). We point out that the operators (1.15) considered in this paper are more general than the operators (1.11) considered in [14]. It is easy to verify when , the operators (1.15) reduce to the operators (1.11). Thus, the operators (1.11) are the special case of the operators (1.15). The literature [14] studied the local approximation and the Voronovskaja-type theorem for the operators (1.11). In the present paper, besides the local approximation of the operators (1.15), our research work is different from the literature in [14]; we study the rate of convergence, the weighted Korovkin-type theorem, and the weighted approximation error of the operators (1.15) and obtain some new results. As regards the operators (1.13) considered in [15], it is obvious that these operators are quite different from the operators (1.15). In [15], for the operators (1.13), a direct approximation result in weighted function space with the help of a weighted Korovkin type theorem on the finite interval was obtained. The weighted approximation error of these operators in terms of weighted modulus of continuity was given. Also, an asymptotic formula was established. In the present paper, we introduce a new weighted modulus of continuity which is different from that in [15]. We obtain some results of the weighted approximation with the help of the new weighted modulus of continuity on the infinite interval.

2. Two Lemmas

For the operators defined by (1.15), we give the following two lemmas.

Lemma 2.1. For , we have (i); (ii); (iii); (iv)   ; (v)    .

Proof. In view of the concepts of the -improper integral and -exponential function, for nonnegative integer , we have (i) When , by the formulas (1.5) and (2.1), we can get (ii)When , using , by the formulas (1.5) and (2.1), we can get (iii)When , using , by the formulas (1.5) and (2.1), we can get (iv)When , by the formulas (1.5) and (2.1), we can get
Using   , we have (v)When , similar to the case of , by simple calculation we can get the desired result.

Lemma 2.2. Let , we have (i); (ii).

Proof. By Lemma 2.1, we have
By Lemma 2.1, we have

3. Local Approximation

Let denote the class of all real valued continuous bounded functions on endowed with the norm . The -functional is defined as where and . By [16, page 177, Theorem 2.4] there exists an absolute constant such that where is the second order modulus of smoothness of . By we denote the usual modulus of continuity of .

Theorem 3.1. Let . For every , we have where is an absolute constant, .

Proof. For , , we define
By Lemma 2.1, we get . Let , , by Taylor’s formula we obtain
By the definition given by (3.6), for , we have
Since , so, by Lemma 2.2, we have
By the definition given by (1.15) and Lemma 2.1, we have
So, by the definition given by (3.6), we obtain
Thus, for , we have
Hence, taking infimum on the right hand side over all , we can get
By inequality (3.2), for every , we have

4. Rate of Convergence

Let be the set of all functions defined on satisfying the condition , where , is a constant depending only on . Let denote the subspace of all continuous functions in . Also let be the subspace of all functions , for which is finite. The norm on is . We denote the modulus of continuity of on the closed interval , by as We observe that for , the modulus of continuity as .

Theorem 4.1. Let , , and be the modulus of continuity of on the finite interval , where . Then we have where , , .

Proof. For and , since , we have .
For and , we have with .
So, for and , we may write
Thus, by Cauchy-Schwartz inequality, we obtain
By Lemma 2.2, for and , we have . Hence, for every , , we obtain
By taking , we immediately get the desired result.

Corollary 4.2. Assume that , . Let , on , where . Then we have where and are given in Theorem 4.1.

Proof. Let , , on . Then for any , we have . So, according to the proof of Theorem 4.1, for , , we have
Using the Hölder inequality with , for any , we get
So, for any , we have The desired result follows immediately.

5. Weighted Approximation

Now we give the weighted approximation result for the operators .

Theorem 5.1. Let the sequence satisfies and as . For , we have

Proof. Using the Theorem in [17], we see that it is sufficient to verify the following three conditions: where .
Since , it is clear that .
By Lemma 2.2, we have
Since and , we have as (see [18]), so, we can obtain .
For , by Lemma 2.1, we have which implies that . In a word, we complete the proof.

It is known that, if is not uniformly continuous on the interval , then the usual first modulus of continuity does not tend to zero as . For every , we would like to take a weighted modulus of continuity which tends to zero as .

For every , let The weighed modulus of continuity was defined by İspir in [19]. It is known that has the following properties.

Lemma 5.2 (see [19]). Let . Then (i) is a monotone increasing function of ; (ii)for each , ; (iii)for each , ; (iv)for each , .

Theorem 5.3. Let , . Then we have the inequality where is a positive constant independent and .

Proof. By the definition of and Lemma 5.2 (iv), for every , we have
Then, we obtain
Also by the Cauchy-Schwartz inequality, we have
Consequently,
In view of Lemmas 2.1 and 2.2, we get
Also,
Now from inequalities (5.10)–(5.12), we have where .
Taking , from the above inequality we can obtain the desired result.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant no. 61170324), the Class A Science and Technology Project of Education Department of Fujian Province of China (Grant no. JA12324), and the Natural Science Foundation of Fujian Province of China (Grant no. 2010J01012). The authors thank the associate editor and the referees for several important comments and suggestions which improve the quality of the paper.

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