Some Approximation Theorems

[摘要] The general theme of this note is illustrated by the following theorem:Theorem 1. Suppose 𝐾 is a compact set in the complex plane and 0 belongs to the boundary 𝜕 𝐾 . Let $mathcal{A}(K)$ denote the space of all functions 𝑓 on 𝐾 such that 𝑓 is holomorphic in a neighborhood of 𝐾 and 𝑓(0) = 0. Also for any given positive integer 𝑚, let $mathcal{A}(m, K)$ denote the space of all 𝑓 such that 𝑓 is holomorphic in a neighborhood of 𝐾 and $f(0) = f'(0) = cdots = f^{(m)}(0) = 0$. Then $mathcal{A}(m, K)$ is dense in $mathcal{A}(K)$ under the supremum norm on 𝐾 provided that there exists a sector $W = {re^{i𝜃}; 0 ≤ r ≤ 𝛿, 𝛼 ≤ 𝜃 ≤ 𝛽}$ such that $W cap K = {0}$. (This is the well-known Poincare's external cone condition).}We present various generalizations of this result in the context of higher dimensions replacing holomorphic with harmonic.

[学科分类] 数学（综合）

[关键词] Good set;sequentially good set;linked component;sequentially good measure;simplicial measure.