Mean Rational Approximation for Compact Subsets with Thin Boundaries

Abstract

In 1991, J. Thomson obtained a celebrated decomposition theorem for $P^t(\hat{I}¼),$ the closed subspace of $L^t(\hat{I}¼)$ spanned by the analytic polynomials, when $1\le t<\ddot{A}\pm .$ In 2008, J. Brennan \cite{b08} generalized Thomson's theorem to $R^t(K,\hat{I}¼),$ the closed subspace of $L^t(\hat{I}¼)$ spanned by the rational functions with poles off a compact subset $K$ containing the support of $\hat{I}¼,$ when the diameters of the components of $\mathbb{C}\setminus K$ are bounded below. We extend the above decomposition theorems for $R^t(K,\hat{I}¼)$ when the boundary of $K$ is not too wild.

arXiv admin note: text overlap with arXiv:1904.06446, arXiv:2212.05392