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
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.
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  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
Lemma 2.2 (see ). If , then
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 . 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  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 , 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 ). 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.
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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