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Rapid Polynomial Approximation on Stein Manifolds

Research paper by Audunn Skuta Snaebjarnarson

Indexed on: 19 Dec '16Published on: 19 Dec '16Published in: arXiv - Mathematics - Complex Variables



Abstract

Let $X$ be a Stein manifold and $\psi:X\to \mathbb{R} \cup \{-\infty \}$ a plurisubharmonic exhaustion function. Denote by $\mathcal{P}_{t\psi}$ the vector space of entire functions $f$ on $X$ satisfying $\log|f(z)|\leq t\psi^+(z)+C$ on $X$ for some constant $C\in \mathbb{R}_+$. Assuming certain curvature restrictions on the $(1,1)$-form $\frac{i}{2}\partial \bar{\partial}e^\psi$ we prove a theorem describing the equivalence between possible holomorphic continuation of a function $f\in \mathcal{O}(K)$ ($K\subset X$ compact) and the decay of the function $t\to d_K(f,\mathcal{P}_{t\psi}),\;t\in \mathbb{R}_+$ as $t\to \infty$ where $d_K(f,\mathcal{P}_{t\psi})$ is the best uniform approximation of $f$ on $K$ by functions from $\mathcal{P}_{t\psi}$. This theorem can be thought of as a generalization of a theorem by Siciak from 1962. We also prove a generalization of Winiarski's theorem from 1971 relating the growth of an entire function $f$ to the decay of the function $t\to d_K(f,\mathcal{P}_{t\psi})$. As a corollary we give sufficient conditions for the space $\mathcal{P}^{\psi}:=\cup_{t\in \mathbb{R}_+}\mathcal{P}_{t\psi}$ to be dense in the Fr\'echet space $\mathcal{O}(X)$ of holomorphic functions on $X$.