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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 613258, 8 pages
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.
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 .
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  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  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 : 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 , 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 , 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 , 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 . 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
Corollary 6. For any and for all , one has uniformly as , for the following pairs of and :(i) and ,(ii) and ,(iii) and .
Corollary 7. Let . Then one has uniformly as , for the following pairs of and :(i) and ,(ii) and ,(iii) and .
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 .
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 .
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.
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