Research Article | Open Access

Jianjun Wang, Chan-Yun Yang, Shukai Duan, "Approximation Order for Multivariate
Durrmeyer Operators with Jacobi Weights", *Abstract and Applied Analysis*, vol. 2011, Article ID 970659, 12 pages, 2011. https://doi.org/10.1155/2011/970659

# Approximation Order for Multivariate Durrmeyer Operators with Jacobi Weights

**Academic Editor:**Pavel Drábek

#### Abstract

Using the equivalence relation between -functional and modulus of smoothness, we establish a strong direct theorem and an inverse theorem of weak type for multivariate Bernstein-Durrmeyer operators with Jacobi weights on a simplex in this paper. We also obtain a characterization for multivariate Bernstein-Durrmeyer operators with Jacobi weights on a simplex. The obtained results not only generalize the corresponding ones for Bernstein-Durrmeyer operators, but also give approximation order of Bernstein-Durrmeyer operators.

#### 1. Introduction

Let be a simplex in defined by
For , we denote by the space of -order Lebesgue integrable functions on with
where denote the space of continuous functions on . For , the multivariate *Bernstein*-*Durrmeyer* Operators with variables on are given by
where and , , with the convention

For multivariate Jacobi weights , (, , , , , . We give some further notations, for , and we write , , ( and

For , the weighted *Sobolev* space is given by
where is the interior of . To characterize the approximation capability of multivariate Bernstein-Durrmeyer operators, we introduce the weighted -functional
and a measure of smoothness of

Since 1967, Durrmeyer introduced Bernstein-Durrmeyer operators, and there are many papers which studied theirproperties [1–7]. In 1991, Zhang studied the characterization of convergence for with Jacobi weights. In 1992, Zhou [5] considered multivariate Bernstein-Durrmeyer operators and obtained a characterization of convergence. In 2002, Xuan et al. studied the equivalent characterization of convergence for with Jacobi weights and obtained the following result.

Theorem 1.1. *For , , the following results are equivalent:*(i)*;
*(ii)*. *

In this paper, using the Ditzian-Totik modulus of smoothness, we will give the upper bound and lower bound of approximation function by on simplex. The main results are as follows.

Theorem 1.2. *If , then
**
And there exists a positive number such that the following inequality is satisfied:
*

Throughout the paper, the letter , appearing in various formulas, denotes a positive constant independent of , , and . Its value may be different at different occurrences, even within the same formula.

From Theorem 1.2, we can easily obtain the following corollary.

Corollary 1.3. *If , , we has the following equivalent results:*(i)*;*(ii)*;*(iii)*.*

#### 2. Some Lemmas

To prove Theorem 1.2, we will show some lemmas in this section. For the simplex , the transformation : defined by satisfies , and is the identity operator. So we have where .

Lemma 2.1. *If , then
*

*Proof. *Letting , , , , , then

Using the transformation , (2.2), (2.4), the method of [7], we can easily get (2.3).

Lemma 2.2 (see [8]). *If , then
*

*Proof. *Lemma 2.2 is proved when in [8]. Similarly, we can prove .

Lemma 2.3. *If , , , is basis function of the classical Bernstein operators, then
*

*Proof. *The first inequality can be inferred by Hölder inequality. In the following we prove the second inequality.(i)If , using Hölder inequality, we can easily obtain the result.(ii)If , let , , , then
Lemma 2.3 is completed.

Lemma 2.4. *If , , then
*

*Proof. *In the following we use the induction on the dimension number to prove the result. The case was proved by Lemma 4 of [6]. Next, suppose that Lemma 2.4 is valid for ; we prove it is also true for . To observe this, we use a decomposition formula (2.4), and we have
where . Thus we have

In the above derivation, we have used the formula [6]
and the inequality
When , we have
where .

From the Cauchy-Swartz inequality, Hölder inequality, and Lemma 2.3, we have
By *Riesz* interpolation theorem, we get
Similarly, the other cases for = can be proved. For , by the transformation , we have
Lemma 2.4 is completed.

Lemma 2.5. *If , , then
*

*Proof. *We use the induction on the dimension number to prove Lemma 2.5. The case was proved by Lemma 3 of [6], that is,
Next, suppose that Lemma 2.5 is valid for , and we prove it is also true for . Noticing formula (2.4), we have
where . When , from the inductive assumption of , we have
When , we have
where . From the inductive assumption, the Cauchy-Swartz inequality, Holder inequality, and Lemma 2.4, we get
By *Riesz* interpolation theorem, we get
Similarly, the other cases for = can be proved. For , by the transformation , we have
Lemma 2.5 is completed.

Lemma 2.6 (see [9]). *Let , be nonnegative sequences (, . For , if the sequences , satisfy
**
one has
**
If , then . If , then .*

#### 3. The Proof of Theorems

Now we prove (1.9) of Theorem 1.2. By using Lemma 2.1, for arbitrary , we have Hence, from Lemma 2.2, we obtain Next, we prove (1.10) of Theorem 1.2. Leting (, , then . By Lemmas 2.4 and 2.5, we have Using Lemma 2.6, we get . That is, When , there exists such that and satisfies the equation Thus, Using Lemma 2.2, we have Theorem 1.2 is completed.

#### Acknowledgments

This paper was supported by the Natural Science Foundation of China (nos. 11001227, 60972155), Natural Science Foundation Project of CQ CSTC (no. CSTC, 2009BB2306, CSTC2009BB2305), and the Fundamental Research Funds for the Central Universities (no. XDJK2010B005, XDJK2010C023).

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#### Copyright

Copyright © 2011 Jianjun Wang et al. 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.