- Feilong Cao
^{1}Email author and - Yongfeng An
^{1}

**2011**:158219

https://doi.org/10.1155/2011/158219

© F. Cao and Y. An. 2011

**Received: **14 November 2010

**Accepted: **17 January 2011

**Published: **13 February 2011

## Abstract

## 1. Introduction

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.

where the infimum is taken over all .

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.

Now we state the main result of this paper.

Theorem 2.2.

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.

which has been proved in [3]

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

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

## Declarations

### Acknowledgment

The research was supported by the National Natural Science Foundation of China (no. 90818020).

## Authors’ Affiliations

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## Copyright

This article is published under license to BioMed Central Ltd. 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.