{
  "id": "1907.04287",
  "version": "v1",
  "published": "2019-07-09T16:48:08.000Z",
  "updated": "2019-07-09T16:48:08.000Z",
  "title": "Approximation in the mean by rational functions II",
  "authors": [
    "Liming Yang"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "For $1\\le t < \\infty$, a compact subset $K\\subset\\mathbb C$, and a finite positive measure $\\mu$ supported on $K$, $R^t(K, \\mu)$ denotes the closure in $L^t(\\mu)$ of rational functions with poles off $K$. Conway and Yang (2019) introduced the concept of non-removable boundary $\\mathcal F$ and removable set $\\mathcal R = K\\setminus \\mathcal F$ for $R^t(K, \\mu)$. We continue the previous work and obtain the decomposition theorem for $R^t(K, \\mu)$. Let $H^\\infty_{\\mathcal R}(A_{\\mathcal R})$ be the weak$^*$ closure in $L^\\infty (A_{\\mathcal R})$ of the functions that are bounded analytic off compact subsets of $\\mathcal F$, where $A_{\\mathcal R}$ denotes the area measure restricted to $\\mathcal R$. We prove: There exists a Borel partion $\\{\\Delta_n\\}_{n\\ge 0}$ of $\\text{spt}(\\mu )$ such that \\[ \\ R^t(K,\\mu) = L^t(\\mu |_{\\Delta_0})\\oplus \\bigoplus_{n=1}^\\infty R^t(K_n, \\mu |_{\\Delta_n}), \\] satisfying, for $n \\ge 1$, (a) $R^t(K_n, \\mu |_{\\Delta_n})$ contains no non-trival characteristic functions; (b) $\\mathcal R_n$ is $\\gamma$-connected, where $\\mathcal R_n$ is the removable set for $R^t(K_n, \\mu |_{\\Delta_n})$; (c) $K_n \\subset \\text{clos}(\\mathcal R_n)$; (d) $K_n \\cap K_m \\subset \\mathcal F$, for $n\\ne m$ ($m \\ge 1$); and (e) there exists an isometric isomorphism and weak$^*$ homeomorphism $\\rho_n$ from $R^t(K_n, \\mu |_{\\Delta_n}) \\cap L^\\infty (\\mu |_{\\Delta_n})$ onto $H^\\infty_{\\mathcal R_n}(A_{\\mathcal R_n })$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-09T16:48:08.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "rational functions",
      "approximation",
      "compact subset",
      "non-trival characteristic functions",
      "removable set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}