Journal of Applied Mathematics

Volume 2013, Article ID 613258, 8 pages

http://dx.doi.org/10.1155/2013/613258

## Quantitative Global Estimates for Generalized Double Szasz-Mirakjan Operators

Eastern Mediterranean University, Gazimagusa, Cyprus, Mersin 10, Turkey

Received 15 December 2012; Accepted 8 May 2013

Academic Editor: Jingxin Zhang

Copyright © 2013 Mehmet Ali Özarslan and Hüseyin Aktuğlu. 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 the generalized double Szász-Mirakjan operators in this paper. We obtain several quantitative estimates for these operators. These estimates help us to determine some function classes (including some Lipschitz-type spaces) which provide uniform convergence on the whole domain .

#### 1. Introduction

The well-known Szász-Mirakjan operators are defined on the space as follows: where is the set of all real functions on such that the right-hand side in (1) make sense for all and . By modifying the Szász-Mirakjan operators as where is a sequence of real-valued, continuous functions defined on with , it has been shown in [1] that if one let then the operators defined by preserve the test function and provide a better error estimation than the operators for all and for each . Note that denotes the space of all bounded and continuous functions on . On the other hand, by letting it has been shown in [2] that the operators defined by do not preserve the test functions and but provide the best error estimation among all the Szász-Mirakjan operators for all and for each . For the other linear positive operator families which preserve , we refer [3–9]. On the other hand, in [10, 11] the authors considered some operators preserving .

Favard was the first to introduce the double Szász-Mirakjan operators [12]: where is the set of all real functions on such that the right-hand side in (7) has a meaning for all and . Recently, Dirik and Demirci have introduced and investigated different variants of the general double Szász-Mirakjan operators: In [13], they considered the case of operators which preserve the test function and provide a better error estimation than the operators for all and for each . On the other hand, in [14], they considered the case Note that for this case, the operators do not preserve any test function (i.e., , , , and ) but provide a better error estimation than the operators for all and .

Finally, we should note that, following the similar arguments as used in [2], the best error estimation among all the general double Szász-Mirakjan operators can be obtained from the case: for all and .

For the operators the following Lemma is straightforward.

Lemma 1. *Let , , , , and. Then, for each and, one has *(a)*,
*(b)*,
*(c)*,
*(d)*. *

#### 2. Global Results

In this section we first introduce the following Lipschitz-type space: where , , is any positive constant, and .

We should note that this space is the bivariate extension of Lipschitz-type space considered earlier by Szasz [15]. For the space with , we have the following approximation result.

Theorem 2. *For any and for each , one has
*

*Proof. *Take . Then, for and for each , we get
Using the Cauchy-Schwarz inequality, we obtain
Secondly let . Then, for and for each , we have
Applying the Hölder inequality with and we have, for any
Hence, the result.

The following lemma will be used in the rest of the paper.

Lemma 3. *One has, for each ,
*

*Proof. * Using the fact that , we get
Finally, applying the Cauchy-Schwarz inequality, we write
Using Lemma 1, we get the result.

Recall that, for all , the modulus of denoted by is defined as

Theorem 4. *Let . Then one has, for each ,
**
where
*

*Proof. *We directly have
Therefore,
Because of the fact that
we have
and hence
Finally, using Lemma 3, the proof is completed.

Theorem 5. *Let . Let
**
where , is any positive constant, and . Then
**
where is the same as in Theorem 4. *

*Proof. *We directly have
Applying the Hölder inequality with and we have
Using Lemma 3, we get the result.

#### 3. Concluding Remarks

In this section we show that taking and or and , in Theorems 2, 4, and 5 gives global results. Also we present the results obtained by Theorems 2, 4, and 5 for and .

Corollary 6. *For any and for all , one has
**
uniformly as , for the following pairs of and :*(i)* and ,*(ii)* and ,*(iii)* and .*

*Proof. *(i) Taking and in (13), we directly have
(ii) Taking and in (13) gives
(iii) Taking and in (13), we have

Corollary 7. *Let . Then one has
**
uniformly as , for the following pairs of and :*(i)* and ,*(ii)* and ,*(iii)* and .*

*Proof. * (i) Taking and in (23), we directly have
(ii) Taking and in (23) gives
where
(iii) Taking and in (23), we have
where

Corollary 8. *Let , and let
**
where , is any positive constant, and . Then
**
uniformly as , for the following pairs of and *(i)* and ,*(ii)* and ,*(iii)* and .*

*Proof. *(i) Taking and in (23), we directly have
(ii) Taking and in (23) gives
where
(iii) Taking and in (23), we have
where

*Remark 9. *Corollaries 7 and 8 conclude that is a real continuous and bounded function on and if is uniformly continuous on , then converges uniformly to as . Note that the one variable version of Corollary 7 was given in [16].

Corollary 10. *Take
**
where Then * (i)* for any , and for each , , one has
* *where
* (ii)* let . Then one has for each * *where
* (iii)* let , and let
* *where , , is any positive constant, and . Then
* *where is the same given in Corollary 10. *

It should be mentioned that, for and , and . Therefore, Corollary 10, Corollary 10, and Corollary 10 reduce to Corollary 6, Corollary 7, and Corollary 8, respectively.

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