Inequalities for a class of multivariate operators

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.


Introduction
Let N be the set of natural numbers, and {s n } ∞ n= (s n ≥ , s n ∈ N) be a sequence. In Furthermore, Cao [] proved that the necessary and sufficient condition of convergence for the operators is lim n→∞ (s n /n) = , and he also proved that for n ∈ Q = {n : n ∈ N, and  < (s n -)/n + / √ n ≤ } the following estimate of approximation degree holds: (  ) http://www.journalofinequalitiesandapplications.com/content/2012/1/175 Here, ω(f , t) 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 (). 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 R d . Let denote the canonical unit vector in R d , i.e., its ith component is  and the others are , and let 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 d = , these operators reduce to the univariate operators defined by (), and when s n =  they are just the well-known multivariate Bernstein operators on the simplex T, B n,d , defined by and Define differential operators: then the weighted Sobolev space can be defined by T is inner of T, and the Peetre K -functional on C(T) is given by

Berens and Xu
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 f ∈ C(T) defined by (see [])

It is shown in [] that
() http://www.journalofinequalitiesandapplications.com/content/2012/1/175 Now we state the main results of this paper as follows.
Theorem . Let f ∈ C(T), then for n ∈ Q = {n : n ∈ N, and  < s n - n +  √ n ≤ }, there holds Theorem . If f ∈ C(T) and lim n→∞ (s n /n) = , then From Theorem . and Theorem ., we easily obtain the following corollaries, which characterize the approximation feature of the multivariate operators L n,d given by ().

Some lemmas
In this section, we prove some lemmas.
Defining the transformation T i (i = , , . . . , d) from T to itself, i.e., we have the following symmetric property for the operators L n,d , which is similar to the known one of the multivariate Bernstein operators (see [, ]).

Lemma . For the above transformation
Proof It is sufficient to prove the case i = . Let The proof of Lemma . is completed.
To prove Theorem ., we need some the following lemmas. At first, similar to the estimates for the Bernstein operators (see [, , ]), it is not difficult to derive the following Lemma .. http://www.journalofinequalitiesandapplications.com/content/2012/1/175 Lemma . The following inequalities hold: Secondly, we need prove two Bernstein type inequalities.
Noting that This inequality shows that Lemma . is valid for d = . For the proof of the case d > , we use a decomposition technique and the induction. In fact, let Therefore, Now, suppose that Lemma . is valid for d -, then from () it follows that So, Lemma . is true for i = . From the symmetry, the proof of the cases i = , , , . . . , d is the same. For the cases  ≤ i < j ≤ d, we use Lemma . and obtain that So, the proof of Lemma . is complete.
Proof We only need to prove the case s n >  because the case s n =  has been shown in []. Our approach is based on the induction. At first, for d = , let h = (n + s n -) - , then by simple calculation we have Let y = (k + j)/(n + s n -), then we have for  ≤ k ≤ n -,  ≤ j ≤ s n -, and for |u| ≤ h, there holds | -y -u| ≤ . Hence, Now, assume that Lemma . is valid for d -, then by () Also, we can check the following inequalities: Thus, Similarly, the cases i = , , , . . . , d can be proved. For the case  ≤ i ≤ j ≤ d, we use the transformation T i and Lemma ., it is easy to verify Hence, the proof of Lemma . 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 . of [] where the proof of case ν  =  and C =  was given.
Lemma . Let μ n , ν n , and ψ n are all nonnegative numerical sequence, and μ  = ν  = . If for  < r < s and  ≤ k ≤ n, n ∈ N, there holds Lemma . Let ν n and ψ n are all nonnegative numerical sequence. If for s >  and  ≤ k ≤ n, n ∈ N, there holds ν n ≤ ( k n ) s ν k + Cψ k , then

The proof of main results
First, we prove Theorem .. By straight calculation, we have (see also []) which implies from Lemma . that μ n ≤ Cn - n k= ψ k , i.e., Let ν n = D i (L n,d f ) and ψ n be the same as the above, then we have by Lemma . ν n ≤ k n ν k + ψ k , ≤ k ≤ n.