Approximation in the mean by rational functions II

TL;DR

This paper explores the structure of rational approximation spaces in complex analysis, focusing on boundary properties and connectivity conditions that influence the functions contained within these spaces.

Contribution

It extends previous work by characterizing when rational approximation spaces lack non-trivial characteristic functions and establishes an isometric isomorphism with a bounded analytic function space.

Findings

R^t(K, μ) contains no non-trivial characteristic functions iff the removable set is γ-connected.

Established an isometric isomorphism between R^t(K, μ)∩L^∞(μ) and a bounded analytic function space.

Proved the equivalence of structural properties under the assumption that S_μ is pure.

Abstract

For $1\le t < \infty$, a compact subset $K\subset\mathbb C$, and a finite positive measure $\mu$ supported on $K$, $R^t(K, \mu)$ denotes the closure in $L^t(\mu)$ of rational functions with poles off $K$. Conway and Yang (2019) introduced the concept of non-removable boundary $\mathcal F$ and removable set $\mathcal R = K\setminus \mathcal F$ for $R^t(K, \mu)$. We continue the previous work and obtain structural results for $R^t(K, \mu)$. Assume that $S_\mu$, the multiplication by $z$ on $R^t(K, \mu)$, is pure ($R^t(K, \mu)$ does not have $L^t$ summand). Let $H^\infty_{\mathcal R}(A_{\mathcal R})$ be the weak$^*$ closure in $L^\infty (A_{\mathcal R})$ of the functions that are bounded analytic off compact subsets of $\mathcal F$, where $A_{\mathcal R}$ denotes the area measure restricted to $\mathcal R$. $\mathcal R$ is $\gamma$-connected ($\gamma$ denotes analytic capacity) if for any two disjoint open set $G_1$ and $G_2$ with $\mathcal R \subset G_1 \cup G_2 ~\gamma-a.a.$, then $\mathcal R \subset G_1 ~\gamma-a.a.$ or $\mathcal R \subset G_2 ~\gamma-a.a.$. We prove: (1) $R^t(K, \mu)$ contains no non-trivial characterization functions if and only if the removable set $\mathcal R$ is $\gamma$-connected. (2) There is an isometric isomorphism and a weak$^*$ homeomorphism from $R^t(K, \mu)\cap L^\infty(\mu )$ onto $H^\infty_{\mathcal R}(A_{\mathcal R })$.