{
  "id": "2005.07372",
  "version": "v1",
  "published": "2020-05-15T06:16:19.000Z",
  "updated": "2020-05-15T06:16:19.000Z",
  "title": "The metric projections onto closed convex cones in a Hilbert space",
  "authors": [
    "Yanqi Qiu",
    "Zipeng Wang"
  ],
  "comment": "30 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "We study the metric projection onto the closed convex cone in a real Hilbert space $\\mathscr{H}$ generated by a sequence $\\mathcal{V} = \\{v_n\\}_{n=0}^\\infty$. The first main result of this paper provides a sufficient condition under which we can identify the closed convex cone generated by $\\mathcal{V}$ with the following set: \\[ \\mathcal{C}[[\\mathcal{V}]]: = \\bigg\\{\\sum_{n=0}^\\infty a_n v_n\\Big|a_n\\geq 0,\\text{ the series }\\sum_{n=0}^\\infty a_n v_n\\text{ converges in $\\mathscr{H}$}\\bigg\\}. \\] Then, by adapting classical results on general convex cones, we give a useful description of the metric projection of a vector onto $\\mathcal{C}[[\\mathcal{V}]]$. As applications, we obtain the best approximations of many concrete functions in $L^2([-1,1])$ by polynomials with non-negative coefficients.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-05-15T06:16:19.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "closed convex cone",
      "metric projection",
      "real hilbert space",
      "general convex cones",
      "first main result"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}