关键词: 广义Bernstein算子, 连续模, 正定理, 逆定理,  , Voronovskaja 型展开

Abstract:

In this paper, the authors introduce and study  generalized Bernstein operators  defined on a simplex in R.
They first prove that these operators possess the symmetric property for a transformation from the simplex to itself and the Lipshcitz property.  Then,  by means of the multivariate Ditzian-Totik's modulus of continuity, they obtain a strong type direct estimate  and an inverse inequality of weak type for the operators approximating multivariate continuous functions. Finally, they  give some Voronovskaja type expansions when the parameters s_n satisfy some conditions. The obtained results include the corresponding ones of the classical Bernstein operators as a corollary.

Key words: Generalized Bernstein operators, Modulus of continuity, Direct theorem, Inverse theorem, Voronovskaja type expansion