多元Bernstein算子的导数与函数的光滑性

DERIVATIVES OF MULTIVARIATE BERNSTEIN OPERATORS AND SMOOTHNESS OF FUNCTION

利用一个新的光滑性度量刻画多元Ｂｅｒｎｓｔｅｉｎ算子方向导数的特征，建立 Ｂｅｒｎｓｔｅｉｎ算子的导数与逼近函数光滑性之间的等价关系．同时，一个关于一元Ｂｅｒｎｓｔｅｉｎ算子的相应结果被推广到多元情形．

We characterize the directional derivatives of multidimensional Bernstein opera- tors by using a new measure of smoothness. This task is carried out by means of establishing a equaivalence relation between the asymptotic behaviour of derivatives of Inultidimensional Bernstein operators and the smoothness of the functions they approximate. Also, a known result on univariate Bernstein operators is extended to the multidimensional cases.