Some approximation theorems

Abstract

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 $\partial K$. Let ${\cal A}(K)$ 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 ${\cal A}(m,K)$ denote the space of all $f$ such that $f$ is holomorphic in a neighborhood of $K$ and $f(0)=f^{\prime}(0)=...=f^{(m)}(0)=0$. Then ${\cal A}(m,K)$ is dense in ${\cal A}(K)$ under the supremum norm on $K$ provided that there exists a sector $W=\{r\hbox{\rm e}^{i\theta}; 0\leq r\leq\delta,\alpha\leq\theta\leq\beta\}$ 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.