Abstract
The main aim of this paper is to introduce and study multivariate Baskakov-Durrmeyer operator, which is nontensor product generalization of the one variable. As a main result, the strong direct inequality of approximation by the operator is established by using a decomposition technique.
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1. Introduction
Let , , . The Baskakov operator defined by
was introduced by Baskakov [1] and can be used to approximate a function defined on . It is the prototype of the Baskakov-Kantorovich operator (see [2]) and the Baskakov-Durrmeyer operator defined by (see [3, 4])
where .
By now, the approximation behavior of the Baskakov-Durrmeyer operator is well understood. It is characterized by the second-order Ditzian-Totik modulus (see [3])
More precisely, for any function defined on , there is a constant such that
where .
Let , which is defined by
Here and in the following, we will use the standard notations
By means of the notations, for a function defined on the multivariate Baskakov operator is defined as (see [5])
where
Naturally, we can modify the multivariate Baskakov operator as multivariate Baskakov-Durrmeyer operator
where
It is a multivariate generalization of the univariate Baskakov-Durrmeyer operators given in (1.2) and can be considered as a tool to approximate the function in .
2. Main Result
We will show a direct inequality of approximation by the Baskakov-Durrmeyer operator given in (1.10). By means of K-functional and modulus of smoothness defined in [5], we will extend (1.4) to the case of higher dimension by using a decomposition technique.
Fox , we define the weight functions
Let
denote the differential operators. For , we define the weighted Sobolev space as follows:
where , , and denotes the interior of . The Peetre -functional on (), are defined by
where the infimum is taken over all .
For any vector in , we write the th forward difference of a function in the direction of as
We then can define the modulus of smoothness of , as
where denotes the unit vector in , that is, its th component is 1 and the others are 0.
In [5], the following result has been proved.
Lemma 2.1.
There exists a positive constant, dependent only on and , such that for any ,
Now we state the main result of this paper.
Theorem 2.2.
If , , then there is a positive constant independent of and such that
Proof.
Our proof is based on an induction argument for the dimension . We will also use a decomposition method of the operator . We report the detailed proof only for two dimensions. The higher dimensional cases are similar.
Our proof depends on Lemma 2.1 and the following estimates:
The first estimate is evident as the are positive and linear contractions on . We can demonstrate the second estimate by reducing it to the one dimensional inequality
which has been proved in [3]
Now we give the following decomposition formula:
where
which can be checked directly and will play an important role in the following proof.
From the decomposition formula, it follows that
where
Then by the Jensen's inequality, we have
However, by definition, one also has
Therefore,
To estimate the second term , we use a similar method as to estimate (2.10) (see [3]) and can get
Denoting , , and , we have
Recalling that is no bigger than or , and the fact
proved in [6] (see [6, Lemma 2.1]), we obtain
and hence
The second inequality of (2.9) has thus been established, and the proof of Theorem 2.2 is finished.
References
Baskakov VA: An instance of a sequence of linear positive operators in the space of continuous functions. Doklady Akademii Nauk SSSR 1957, 113: 249–251.
Ditzian Z, Totik V: Moduli of Smoothness, Springer Series in Computational Mathematics. Volume 9. Springer, New York, NY, USA; 1987:x+227.
Heilmann M: Direct and converse results for operators of Baskakov-Durrmeyer type. Approximation Theory and its Applications 1989,5(1):105–127.
Sahai A, Prasad G: On simultaneous approximation by modified Lupas operators. Journal of Approximation Theory 1985,45(2):122–128. 10.1016/0021-9045(85)90039-5
Cao F, Ding C, Xu Z: On multivariate Baskakov operator. Journal of Mathematical Analysis and Applications 2005,307(1):274–291. 10.1016/j.jmaa.2004.10.061
Chen W, Ditzian Z: Mixed and directional derivatives. Proceedings of the American Mathematical Society 1990,108(1):177–185. 10.1090/S0002-9939-1990-0994773-0
Acknowledgment
The research was supported by the National Natural Science Foundation of China (no. 90818020).
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Cao, F., An, Y. Approximation by Multivariate Baskakov-Durrmeyer Operator. J Inequal Appl 2011, 158219 (2011). https://doi.org/10.1155/2011/158219
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DOI: https://doi.org/10.1155/2011/158219