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arXiv math/0312123math.FAmath.CV
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几个逼近定理

Some approximation theorems

N. V. Rao

备注：
4 pages, no figures, no tables
发表信息：
Proc. Indian Acad. Sci. (Math. Sci.), Vol. 113, No. 1, February 2003, pp. 87-90
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摘要：

本说明的一般主题由以下定理说明：定理1。设$K$是复平面上的紧集，0属于边界$\partial K$。令${\cal A}(K)$表示$K$上的所有函数$f$的空间，使得$f$在$K$和$f(0)=0$的邻域中是全纯的。同样，对于任何给定的正整数$m$，令${\cal A}(m,K)$表示所有$f$的空间，使得$f$在$K$和$f(0)=f^{\prime}(0)=...=f^{(m)}(0)=0$的邻域中是全纯的。如果存在一个扇区${\cal A}(m,K)$使得$K$. (This是著名的庞加莱外锥条件）。我们提出了各种概括的背景下，更高的维度取代全纯调和的这一结果。

英文摘要：

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 $\partial K$. 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 $W=\{r\hbox{\rm e}^{iθ}; 0\leq r\leqδ,α\leqθ\leqβ\}$ 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.

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