{
  "id": "2212.10811",
  "version": "v1",
  "published": "2022-12-21T06:58:48.000Z",
  "updated": "2022-12-21T06:58:48.000Z",
  "title": "Mean Rational Approximation for Compact Subsets with Thin Boundaries",
  "authors": [
    "John B. Conway",
    "Liming Yang"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:1904.06446, arXiv:2212.05392",
  "categories": [
    "math.FA"
  ],
  "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 holds",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-12-21T06:58:48.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "mean rational approximation",
      "compact subset",
      "thin boundaries",
      "decomposition theorem holds",
      "closed subspace"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}