Approximation by Multivariate Baskakov-Durrmeyer Operator

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.

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])

(13)

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

(14)

(15)

where .

Let , which is defined by

(16)

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

(17)

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

(18)

where

(19)

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

(110)

where

(111)

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

(21)

Let

(22)

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

(23)

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

(24)

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

(25)

We then can define the modulus of smoothness of , as

(26)

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

(28)

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

(210)

which has been proved in [3]

Now we give the following decomposition formula:

(211)

where

(212)

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

From the decomposition formula, it follows that

(213)

where

(214)

Then by the Jensen's inequality, we have

(215)

However, by definition, one also has

(216)

Therefore,

(217)

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

(218)

Denoting , , and , we have

(219)

Recalling that is no bigger than or , and the fact

(220)

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

(221)

and hence

(222)

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