关于局部饱和问题的一点注记

A NOTE ON LOCAL SATURATION

设{Ln}是从 C[a,b]到 C[c,d]的一列算子,[c,d][a,b],如果存在一个函数列{φn（x）}在[c,d]上一致趋于0,在（c,d）上为正,满足以下两条:（1）存在函数类 T（Ln）使（φn（x））-1[f（x）-Ln（f,x）]=0,x∈（c,d）,成立,当且仅当 f∈T（Ln）.（2）存在函数 fn∈C[a,b],f0∈T（Ln）。

Theorem 1.Let{L_n}be a sequence of positive linear operators from C[a,b]into C[c,d]with a<c<d<b and 0<μ_n=‖L_n(t-x,x)‖[c,d]→0. Suppose that {L_n} satisfies the following three conditions: (1)‖1-L_n(1,x)‖=o(μ_n). (2)L_n((t-x)~2,x)=o(μ_n),x(c,d). (3)There are c_1,c_2>0 such that c_1μ_n|≤L_n(t-x,x)|≤c_2μ_n,x(c,d)and sign L_n(t-x,x)is constant on(c,d). Then {L_n} is saturated on [c,d] with order μ_n,the trivial class T(L_n)={f; f is constant on [c,d]} and the saturation class S(L_n)={f;fLip 1 on [c,d]}.