关于多元多项式凸逼近

§1.引言 在多项式保形逼近理论中面临以下两个基本问题: 问题1.对于k≥2,R~k中是否存在k维紧凸集E及C(E)上的保凸正线性算子列L_n:C(E)→P_n满足:?凸函数f∈C(E),||L_nf-f||_E→0(n→∞)? 问题2.对于k≥2以及R~k中的任意k维紧凸集E和任意凸函数f∈C(E),是否存在一列多项式p_n∈P_n,使每一个p_n在E上为凸函数,并且||p_n-f||_E→0(n→∞)?

ON MULTIVARIATE POLYNOMIALS CONVEX APPROXIMATION

The main purpose of this paper is to establish a theorem on multivariate polynomial convexapproximation. It is also pointed out that the positive linear operator which preserves conve-xity has relation to its behaviour on linear functions.