{
  "id": "1905.02516",
  "version": "v1",
  "published": "2019-05-07T12:58:27.000Z",
  "updated": "2019-05-07T12:58:27.000Z",
  "title": "On $L_2$-approximation in Hilbert spaces using function values",
  "authors": [
    "David Krieg",
    "Mario Ullrich"
  ],
  "comment": "9 pages",
  "categories": [
    "math.NA",
    "math.PR"
  ],
  "abstract": "We study $L_2$-approximation of functions from Hilbert spaces $H$ in which function evaluation is a continuous linear functional, using function values as information. Under certain assumptions on $H$, we prove that the $n$-th minimal worst-case error $e_n$ satisfies \\[ e_n \\,\\lesssim\\, a_{n/\\log(n)}, \\] where $a_n$ is the $n$-th minimal worst-case error for algorithms using arbitrary linear information, i.e., the $n$-th approximation number. Our result applies, in particular, to Sobolev spaces with dominating mixed smoothness $H=H^s_{\\rm mix}(\\mathbb{T}^d)$ with $s>1/2$ and we obtain \\[ e_n \\,\\lesssim\\, n^{-s} \\log^{sd}(n). \\] This improves upon previous bounds whenever $d>2s+1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-05-07T12:58:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "41A25",
      "41A46",
      "60B20",
      "41A63",
      "46E35"
    ],
    "keywords": [
      "hilbert spaces",
      "function values",
      "th minimal worst-case error",
      "arbitrary linear information",
      "th approximation number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}