In 1991, J. Thomson obtained a celebrated decomposition theorem for
Pt(μ), the closed subspace of Lt(μ) spanned by the analytic
polynomials, when 1 \le t < \i. In 2008, J. Brennan \cite{b08} generalized
Thomson's theorem to Rt(K,μ), the closed subspace of Lt(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 C∖K are bounded below. We obtain a necessary and sufficient condition for
Rt(K,μ) to ensure such a decomposition theorem holdsComment: arXiv admin note: text overlap with arXiv:1904.06446,
arXiv:2212.0539