Mean Rational Approximation for Compact Subsets with Thin Boundaries

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 obtain a necessary and sufficient condition for $R^t(K, \mu)$ to ensure such a decomposition theorem holdsComment: arXiv admin note: text overlap with arXiv:1904.06446, arXiv:2212.0539