Mean Rational Approximation for Compact Subsets with Thin Boundaries

Abstract

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