Approximation in the mean by rational functions II

Liming Yang

Published 2019-07-09Version 1

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 the decomposition theorem for $R^t(K, \mu)$. 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 $\mathcal R$. We prove: There exists a Borel partion $\{\Delta_n\}_{n\ge 0}$ of $\text{spt}(\mu )$ such that \[ \ R^t(K,\mu) = L^t(\mu |_{\Delta_0})\oplus \bigoplus_{n=1}^\infty R^t(K_n, \mu |_{\Delta_n}), \] satisfying, for $n \ge 1$, (a) $R^t(K_n, \mu |_{\Delta_n})$ contains no non-trival characteristic functions; (b) $\mathcal R_n$ is $\gamma$-connected, where $\mathcal R_n$ is the removable set for $R^t(K_n, \mu |_{\Delta_n})$; (c) $K_n \subset \text{clos}(\mathcal R_n)$; (d) $K_n \cap K_m \subset \mathcal F$, for $n\ne m$ ($m \ge 1$); and (e) there exists an isometric isomorphism and weak$^*$ homeomorphism $\rho_n$ from $R^t(K_n, \mu |_{\Delta_n}) \cap L^\infty (\mu |_{\Delta_n})$ onto $H^\infty_{\mathcal R_n}(A_{\mathcal R_n })$.