Mean Rational Approximation for Compact Subsets with Thin Boundaries

TL;DR

This paper extends decomposition theorems for rational approximation spaces on compact sets with less regular boundaries, broadening the applicability of earlier results by Thomson and Brennan.

Contribution

It generalizes existing decomposition theorems for rational approximation spaces to include compact sets with less regular, 'not too wild' boundaries.

Findings

Extended decomposition theorems to more general compact sets.

Broadened understanding of rational approximation on irregular boundaries.

Provided conditions under which the theorems hold.

Abstract

In 1991, J. Thomson obtained a celebrated decomposition theorem for $P^t(\mu),$ the closed subspace of $L^t(\mu)$ spanned by the analytic polynomials, when $1 \le t < \i.$ In 2008, J. Brennan \cite{b08} generalized Thomson's theorem to $R^t(K, \mu),$ the closed subspace of $L^t(\mu)$ spanned by the rational functions with poles off a compact subset $K$ containing the support of $\mu,$ when the diameters of the components of $\mathbb C\setminus K$ are bounded below. We extend the above decomposition theorems for $R^t(K, \mu)$ when the boundary of $K$ is not too wild.