This paper introduces and studies a class of generalized multivariate Bernstein operators defined on the simplex. By means of the modulus of continuity and so-called Ditzian-Totik’s modulus of function, the direct and inverse inequalities for the operators approximating multivariate continuous functions are simultaneously established. From these inequalities, the characterization of approximation of the operators follows. The obtained results include the corresponding ones of the classical Bernstein operators.
MSC: 41A25, 41A36, 41A60, 41A63.
Keywords:generalized Bernstein operators; direct and inverse inequalities; characterization of approximation
Let be the set of natural numbers, and ( , ) be a sequence. In , Cao introduced the following generalized Bernstein operators defined on :
where , , and
Clearly, when , reduce to the classical Bernstein operators, , given by
Furthermore, Cao  proved that the necessary and sufficient condition of convergence for the operators is , and he also proved that for the following estimate of approximation degree holds:
Here, is the modulus of continuity of first order of the function f. In , some approximation properties for the operators were further investigated.
In this paper, we will introduce and study the multivariate version defined on the simplex of the generalized Bernstein operators given by (1). The main aim is to establish the direct and inverse inequalities of approximation, which will imply the characterization of approximation of the operators.
For convenience, we denote by bold letter the vector in . Let
denote the canonical unit vector in , i.e., its ith component is 1 and the others are 0, and let
be the simplex in . For , , we denote as usual
Then the well-known Bernstein basis function on T is given by
By means of the basis function, we define the multivariate generalized Bernstein operators on the simplex T as
Obviously, when , these operators reduce to the univariate operators defined by (1), and when they are just the well-known multivariate Bernstein operators on the simplex T, , defined by
Let denote the space of continuous functions on T with the norm defined by , . For arbitrary vector , we write for the rth symmetric difference of a function f in the direction of e
Then the Ditzian-Totik’s modulus of function is defined by (see )
where the weighted functions
Define differential operators:
then the weighted Sobolev space can be defined by
where is inner of T, and the Peetre K-functional on is given by
Berens and Xu  proved that is equivalent to , i.e.,
here and in the following C denotes a positive constant independent of f and n, but its value may be different at a different occurrence.
We also need the usual modulus of continuity of function defined by (see )
where and , and another -functional given by (see )
It is shown in  that
Now we state the main results of this paper as follows.
Theorem 1.1Let , then for , there holds
Theorem 1.2If and , then
Theorem 1.3If and , then there hold
From Theorem 1.2 and Theorem 1.3, we easily obtain the following corollaries, which characterize the approximation feature of the multivariate operators given by (8).
Corollary 1.1Let , . Then, for the Bernstein operators given by (9), the necessary and sufficient condition for which
is ( ).
Corollary 1.2If , , and , then and imply
Corollary 1.3If and ( ), , then for any and , the statement
implies that and ( ).
From Corollary 1.2 and Corollary 1.3, we have the following.
Corollary 1.4Let and ( ), then, for any the necessary and sufficient condition for which
is and ( ).
2 Some lemmas
In this section, we prove some lemmas.
Defining the transformation ( ) from T to itself, i.e.,
Lemma 2.1For the above transformation , , there holds
where , .
Proof It is sufficient to prove the case . Let
and , . Then, from definition (8), it follows that
The proof of Lemma 2.1 is completed. □
Lemma 2.2The following inequalities hold:
Secondly, we need prove two Bernstein type inequalities.
Lemma 2.3Let , . Then
Proof For , by direct computation we have (see )
This inequality shows that Lemma 2.3 is valid for . For the proof of the case , we use a decomposition technique and the induction. In fact, let
then we can decompose the generalized Bernstein operators as
Now, suppose that Lemma 2.3 is valid for , then from (12) it follows that
So, Lemma 2.3 is true for . From the symmetry, the proof of the cases is the same. For the cases , we use Lemma 2.1 and obtain that
So, the proof of Lemma 2.3 is complete. □
Lemma 2.4For , , one has
Proof We only need to prove the case because the case has been shown in . Our approach is based on the induction. At first, for , let , then by simple calculation we have
Let , then we have for , ,
and for , there holds . Hence,
Now, assume that Lemma 2.4 is valid for , then by (12)
Also, we can check the following inequalities:
Similarly, the cases can be proved. For the case , we use the transformation and Lemma 2.1, it is easy to verify
Hence, the proof of Lemma 2.4 is complete. □
We also need the following two interesting results related to nonnegative numerical sequence. The proof of the first result can be found in , and the proof of the other is similar to Lemma 2.1 of  where the proof of case and was given.
Lemma 2.5Let , , and are all nonnegative numerical sequence, and . If for and , , there holds
Lemma 2.6Let and are all nonnegative numerical sequence. If for and , , there holds , then
3 The proof of main results
First, we prove Theorem 1.1. By straight calculation, we have (see also )
Then we use the same method as Theorem 2 of  and obtain easily
We now prove Theorem 1.2. We use a known estimation on Bernstein operators (see ) as an intermediate step to deduce the direct theorem. Since
from the fact that (see )
This completes the proof of Theorem 1.2.
Finally, we prove Theorem 1.3. Let
and , then and from Lemma 2.2, Lemma 2.3, and Lemma 2.4, we have for
which implies from Lemma 2.5 that , i.e.,
Let and be the same as the above, then we have by Lemma 2.2
So, using Lemma 2.6 gives , namely,
For , there is an , such that and hold for . Then
So, combining (10), (13), and (15) we see
Also, collecting (11), (13), and (15) implies
The proof of Theorem 1.3 is complete.
The authors declare that they have no competing interests.
This research was supported by the National Nature Science Foundation of China (No. 61101240, 61272023), the Zhejiang Provincial Natural Science Foundation of China (No. Y6110117), and the Science Foundation of the Zhejiang Education Office (No. Y201122002).
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