多项式的全纯近似,其指数限制在凸锥范围内

Álfheiður Edda Sigurðardóttir
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摘要

我们通过多项式的适当子环来研究几个复变数的全纯函数的近似。这些子环由多个复变函数的多项式组成,这些多项式的指数被限制在某个紧凑凸$S\in\mathbb{R}^n_+$的指定凸锥$\mathbb{R}_+S$内。与集合的多项式全域类似,我们可以定义一个集合 $K$ 相对于给定环的全域,这里用 $\widehat K{}^S$ 表示。通过研究下极值函数 $V^S_K(z)$,我们展示了这些子环在满足 $K= \widehat K{}^S$ 和 $V^{S*}_K|_K=0$ 的 $\mathbb{C}^{*n}$ 紧凑子集上的 Runge-Oka-Weil Theoremon 近似的一个版本。我们为紧凑的莱因哈特集合 $K$ 证明了一个更尖锐的结果:当且仅当全态函数在 $\widehat K{}^S$ 上有界时,全态函数在 $\widehat K{}^S$ 上是可以由环的成员均匀逼近的。我们还证明,如果 $K$ 是 $\mathbb{C}^{*n}$ 的紧凑莱因哈特子集,那么我们有 $V^S_K(z)=\sup_{s\in S}.(\langle s,{\operatorname{Log}\, z}\rangle- \varphi_A(s)) $,其中 $\varphi_A$ 是 $A=\operatorname{Log}\, K=\{(\log|z_1|,\dots,\log|z_n|) \,;\, z\in K\}$ 的支持函数。
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Holomorphic approximation by polynomials with exponents restricted to a convex cone
We study approximations of holomorphic functions of several complex variables by proper subrings of the polynomials. The subrings in question consist of polynomials of several complex variables whose exponents are restricted to a prescribed convex cone $\mathbb{R}_+S$ for some compact convex $S\in \mathbb{R}^n_+$. Analogous to the polynomial hull of a set, we denote the hull of $K$ with respect to the given ring by can define hulls of a set $K$ with respect to the given ring, here denoted $\widehat K{}^S$. By studying an extremal function $V^S_K(z)$, we show a version of the Runge-Oka-Weil Theorem on approximation by these subrings on compact subsets of $\mathbb{C}^{*n}$ that satisfy $K= \widehat K{}^S$ and $V^{S*}_K|_K=0$. We show a sharper result for compact Reinhardt sets $K$, that a holomorphic function is uniformly approximable on $\widehat K{}^S$ by members of the ring if and only if it is bounded on $\widehat K{}^S$. We also show that if $K$ is a compact Reinhardt subsets of $\mathbb{C}^{*n}$, then we have $V^S_K(z)=\sup_{s\in S} (\langle s ,{\operatorname{Log}\, z}\rangle- \varphi_A(s)) $, where $\varphi_A$ is the supporting function of $A=\operatorname{Log}\, K= \{(\log|z_1|,\dots, \log|z_n|) \,;\, z\in K\}$.
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