本文研究单纯形上多元Stancu多项式与连续模之间的关系,证明了Stancu多项式具有保持连续模的性质,推广了一元Bernstein多项式的相应结果.同时,利用多元函数的Ditzian-Totik连续模估计Stancu多项式逼近多元连续函数速度的上界和下界,得到一个使得逼近速度为O(n-a)(0

This paper investigates the relationship between the multivariate Stancu polynomials and the modulus of continuity. First, it is shown that the property of modulus of continuity is preserved by the multivariate Stancu polynomials and hence a corresponding one to univariate Bernstein polynomials is generalized. Then, by means of the Ditzian-Totik's modulus of continuity, the estimates of upper and lower bounds of rate for the polynomials approximating the continuous function are given. A necessary and sufficient condition on the smoothness of function is also obtained from the estimates so that the rate of approximation is O(n-a) (0 < a < 1).