{
  "id": "1904.06446",
  "version": "v1",
  "published": "2019-04-12T23:09:22.000Z",
  "updated": "2019-04-12T23:09:22.000Z",
  "title": "Approximation in the mean by rational functions",
  "authors": [
    "John B. Conway",
    "Liming Yang"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "For $1\\le t < \\infty$, a compact subset $K$ of the complex plane $\\mathbb C$, and a finite positive Borel measure $\\mu$ supported on $K$, $R^t(K, \\mu)$ denotes the closure in $L^t(\\mu)$ of the rational functions with poles off $K$. Let $\\text{abpe}(R^t(K, \\mu))$ denote the set of analytic bounded point evaluations for $R^t(K, \\mu)$. The purpose of this paper is to describe the structure of $R^t(K, \\mu)$. In the work of Thomson on describing the closure in $L^t(\\mu)$ of analytic polynomials, $P^t(\\mu)$, the existence of analytic bounded point evaluations plays critical roles, while $\\text{abpe}(R^t(K, \\mu))$ may be empty. We introduce the concept of non-removable boundary $\\mathcal F$ for $R^t(K, \\mu)$ such that $\\mathcal R = K\\setminus \\mathcal F$ contains $\\text{abpe}(R^t(K, \\mu))$. The sets $\\mathcal F$ and $\\mathcal R$ are essential for us to describe $R^t(K, \\mu)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-04-12T23:09:22.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "rational functions",
      "approximation",
      "point evaluations plays critical roles",
      "analytic bounded point evaluations plays",
      "finite positive borel measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}