Derivatives of Multivariate Bernstein Operators and Smoothness with Jacobi Weights

Using the modulus of smoothness, directional derivatives of multivariate Bernstein operators with weights are characterized. The obtained results partly generalize the corresponding ones for multivariate Bernstein operators without weights.


Introduction
For the simplex S S d in R d d 1, 2, . . .,

S
x x 1 , x 2 , . . ., x d ; x i ≥ 0, i 1, 2, . . ., d, |x| we denote C S the space of continuous functions on S equipped with the norm 1.5 Obviously, the multivariate Bernstein operators given in 1.3 can be reduced as the classical Bernstein polynomials in case d 1, that is, Here introduce the crucial notations of our investigation.First, with the simplex S, we denote V S the set of unit vectors in the directions of the edges of S where e i and −e i are considered to be the same vectors.That is, e i 0, 0, . . ., where d x, y is the Euclidean distance between x and y in R d .Obviously, as x ∈ S, the ϕ 2 ξ x can further be expressed as: It is clear that ϕ 2 ξ x can be reduced as the classical Bernstein polynomials' step-weight function ϕ 2 x ϕ ξ x 2 x 1 − x x ∈ 0, 1 in case d 1.The multivariate Jacobi weight function in this paper is denoted as follows: The rth symmetric difference of function f with the direction e is given by

1.10
Using the above notation, the weighted Sobolev space in S is then defined by 11 where • S is the inner of S. Furthermore, the weighted K-functional is defined by and the weighted modulus is where ωf max x∈S |ω x f x | is the weighted form.From 1 , there exists a positive constant C, 1.14 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.
The close connection between the derivatives of Bernstein-type operators and the smoothness of functions has been well investigated by Ditzian, Totik, Ivanov and some other mathematicians see 2-6 , etc.In 2 , Ditzian has studied the relations between the derivatives of classical Bernstein operators B n,1 f, x and the smoothness of the function f.In 7 , we have presented the relation between the derivatives of classical Bernstein operators and the smoothness of function f with Jacobi weights.Zhou has considered the approximation problems of higher-dimensional Bernstein operators with Jacobi weights, and has pointed out the unboundedness of Bernstein operators with Jacobi weights in the usual norm 8 .Because of the unboundedness of B n,d f, x operators with weights in C S , he used the method of space reduction, that is, has been taken instead of C S ∂S is the boundary of S .He then has shown the characteristic of the two dimensional Bernstein operators with Jacobi weights.In 1 , Cao has yielded the order of approximation of d-dimensional Bernstein Operators with Jacobi weights by using the equivalence relation 1.14 .In 6 , Cao has evaluated extensively derivatives of the multivariate Bernstein operators on a simplex, and he proved the following.
In this paper, we study the characterization of derivatives of multivariate Bernstein polynomials with Jacobi weights by using the measure of smoothness in the space C 0 S .The main result is expressed as follows.
Theorem 1.2.Let f ∈ C 0 S , 0 < α ≤ r, r ∈ N, and ξ ∈ V S , and suppose 1.17 Remark 1.3.Theorem 1.2 shows that the characterization of derivatives for multivariate bernstein operators with jacobi weight by using the measure of smoothness Ω ξ r f, t ω .conversely, we conjecture that the inverse theorem is also correct, that is, The above equivalent relation without Jacobi weight has been proved in 6 when λ 1.In fact, the proof of Theorem 1.2 shows that the direct part holds true, we leave the inverse part as an open problem.

Lemmas
To prove Theorem 1.2, some lemmas will be shown in this section.

2.2
Consider different conditions, By the same methods J ≤ C 1 − x −2β can also be given.Suppose the lemma is correct when d − 1.We prove the lemma is also correct when d. Through a simple computation, the following results can be easily obtained where Journal of Applied Mathematics Lemma 2.2.Letf ∈ C 0 S , r ∈ N, and ξ ∈ V S , then Proof.First, we recall the discussion of theorem 4.1 of 9 that will allow us to consider lemma 1 with ξ e 2 .it is clear that if ξ e i , i 1, 3, 4, . . ., d, we may just rename the coordinates.the following transformation will help us to complete the other case of ξ. the transformation T : S → S is defined by 9 where u i x i i / j ; u j 1 − |x| and I is the identity operator.Obviously, where f T u f x and u T x.So, for ξ e ij / √ 2, 1 ≤ i < j ≤ d, we have

2.11
Secondly, we prove B n,d f ≤ Cn r ωf .

2.12
In The following we use mathematical induction on the dimension number d to prove 2.12 .When d 1, Lemma 3.2 in 10 proved the above inequality for r 1, for r > 1, from the expression of derivatives of Bernstein operator in 4 page125, 9.4.3 , we can easily prove it.Next, suppose that 2.12 is valid for d − 1 d > 1 ; we prove 2.12 is also true for d.Assume ω x can therefore be rewritten as and B n,d f, x can be decomposed as where H u f k 1 /n, 1 − k 1 /n u .Using the inductive assumption, we have

2.16
Here, the equality 2.17 and the inequality f .

2.21
The steps to prove 2.21 are similar to those to prove the inequality 2.12 .Hence, the proof of Lemma 2.3 is complete

Proof of Theorem
We will prove Theorem 1.2 in the followings.For ξ e 2 and for all g ∈ W r
From the definition of K-functional and 1.14 , we obtain Similarly, the case of ξ e i , i 1, 3, 4, ..., d can also be proved.Ifξ e i − e j / √ 2 1 ≤ i < j ≤ d, it is not difficult to obtain Cn r−α , 3.3by assuming η e i , u Tx.The proof of Theorem 1.2 is complete.
2rB n,d f T , u ≤