Approximation Order for Multivariate Durrmeyer Operators with Jacobi Weights

and Applied Analysis 3 Since 1967, Durrmeyer introduced Bernstein-Durrmeyer operators, and there are many paperswhich studied theirproperties 1–7 . In 1991, Zhang studied the characterization of convergence forMn,1 f ;x with Jacobi weights. In 1992, Zhou 5 considered multivariate Bernstein-Durrmeyer operators Mn,d f ;x and obtained a characterization of convergence. In 2002, Xuan et al. studied the equivalent characterization of convergence for Mn,d f ;x with Jacobi weights and obtained the following result. Theorem 1.1. For ωf ∈ L S , 0 < r < 1, the following results are equivalent: i ‖ω Mn,df − f ‖p O n−r ; ii K2 φ f, t ω O t r . In this paper, using the Ditzian-Totik modulus of smoothness, we will give the upper bound and lower bound of approximation function by Mn,d f ;x on simplex. The main results are as follows. Theorem 1.2. If ωf ∈ L S , then ∥∥ω Mn,df − f ∥∥p ≤ C { ω2 φ ( f, 1 √ n )


Introduction
Let S S d d 1, 2, . . .be a simplex in R d defined by x i ≤ 1 .

1.1
For p ≥ 1, we denote by L p S the space of p-order Lebesgue integrable functions on S with For multivariate Jacobi weights ω x We give some further notations, for x ∈ S, and we write ϕ i x ϕ ii x 1.5 For f ∈ L p S , the weighted Sobolev space is given by In this paper, using the Ditzian-Totik modulus of smoothness, we will give the upper bound and lower bound of approximation function by M n,d f; x on simplex.The main results are as follows.
And there exists a positive number δ 0 < δ < 1 such that the following inequality is satisfied: Throughout the paper, the letter C, appearing in various formulas, denotes a positive constant independent of n, x, and f.Its value may be different at different occurrences, even within the same formula.
From Theorem 1.2, we can easily obtain the following corollary.

Some Lemmas
To prove Theorem 1.2, we will show some lemmas in this section.For the simplex S, the transformation T: S → S 10 defined by satisfies T 2 I, and I is the identity operator.So we have

2.4
Using the transformation T, 2.2 , 2.4 , the method of 7 , we can easily get 2.3 .
Proof.Lemma 2.2 is proved when f ∈ C S in 8 .Similarly, we can prove f ∈ L p S .

2.6
Proof.The first inequality can be inferred by Hölder inequality.In the following we prove the second inequality.i If 0 < b < 1, using H ölder inequality, we can easily obtain the result. ii Proof.In the following we use the induction on the dimension number d to prove the result.The case d 1 was proved by Lemma 4 of 6 .Next, suppose that Lemma 2.4 is valid for d r r ≥ 1 ; we prove it is also true for d r 1.To observe this, we use a decomposition formula 2.4 , and we have Cn ωf 1 .

2.10
In the above derivation, we have used the formula 6

2.12
When p ∞, we have From the Cauchy-Swartz inequality, H ölder inequality, and Lemma 2.3, we have

2.14
By Riesz interpolation theorem, we get

2.15
Similarly, the other cases for i 1, 3, 4, . . ., d j can be proved.For i / j, by the transformation T, we have

2.17
Proof.We use the induction on the dimension number d to prove Lemma 2.5.The case d 1 was proved by Lemma 3 of 6 , that is,

2.18
Next, suppose that Lemma 2.5 is valid for d r r ≥ 1 , and we prove it is also true for d r 1. Noticing formula 2.4 , we have When p 1, from the inductive assumption of p 1, we have

2.20
When p ∞, we have From the inductive assumption, the Cauchy-Swartz inequality, Holder inequality, and Lemma 2.4, we get

2.22
By Riesz interpolation theorem, we get

2.23
Similarly, the other cases for i 1, 3, 4, . . ., d j can be proved.For i / j, by the transformation T, we have Lemma 2.5 is completed.

The Proof of Theorems
Now we prove 1.9 of Theorem 1.2.By using Lemma 2.1, for arbitrary g ∈ W r,p φ S ⊂ L p S , we have

S x x 1
, x 2 , . . ., x d : x i ≥ 0, i 1, 2, . . ., d, |x| d i 0 .2 where L ∞ S C S denote the space of continuous functions on S. For f ∈ L S , the multivariate Bernstein-Durrmeyer Operators with d variables on S are given by − |k| !x k 1 − |x| n−|k| x ∈ S and x x 1 , x 2 , . . ., x d ∈ R d , k k 1 , k 2 , . . ., k d ∈ N d 0 , with the convention 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 M n,1 f; x with Jacobi weights.In 1992, Zhou 5 considered multivariate Bernstein-Durrmeyer operators M n,d f; x and obtained a characterization of convergence.In 2002, Xuan et al. studied the equivalent characterization of convergence for M n,d f; x with Jacobi weights and obtained the following result.
Next, we prove 1.10 of Theorem 1.2.Letingσ n C 1/n ωϕ 2 ij D 2 ij M n,d f C ω M n,d f − f p ,then σ 1 0. By Lemmas 2.4 and 2.5, we have When n ≥ 2, there exists m ∈ N such that n/2 ≤ m ≤ n and satisfies the equation p 1 ≤ i ≤ j ≤ d , φ n n ≤ C 1/n n k 1 n/k δ φ k 0 < δ < 1 .That is,