1. Introduction

Let , , . The Baskakov operator defined by

(11)

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])

(12)

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])

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More precisely, for any function defined on , there is a constant such that

(14)
(15)

where .

Let , which is defined by

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Here and in the following, we will use the standard notations

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By means of the notations, for a function defined on the multivariate Baskakov operator is defined as (see [5])

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where

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Naturally, we can modify the multivariate Baskakov operator as multivariate Baskakov-Durrmeyer operator

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where

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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

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Let

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denote the differential operators. For , we define the weighted Sobolev space as follows:

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where , , and denotes the interior of . The Peetre -functional on (), are defined by

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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

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We then can define the modulus of smoothness of , as

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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 ,

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Now we state the main result of this paper.

Theorem 2.2.

If , , then there is a positive constant independent of and such that

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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:

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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

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which has been proved in [3]

Now we give the following decomposition formula:

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where

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which can be checked directly and will play an important role in the following proof.

From the decomposition formula, it follows that

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where

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Then by the Jensen's inequality, we have

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However, by definition, one also has

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Therefore,

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To estimate the second term , we use a similar method as to estimate (2.10) (see [3]) and can get

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Denoting , , and , we have

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Recalling that is no bigger than or , and the fact

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proved in [6] (see [6, Lemma 2.1]), we obtain

(221)

and hence

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The second inequality of (2.9) has thus been established, and the proof of Theorem 2.2 is finished.