{
  "id": "1004.5289",
  "version": "v2",
  "published": "2010-04-29T13:52:21.000Z",
  "updated": "2010-05-19T09:00:30.000Z",
  "title": "Spline approximation of a random process with singularity",
  "authors": [
    "Konrad Abramowicz",
    "Oleg Seleznjev"
  ],
  "comment": "16 pages, 2 figure typos and references corrected, revised classes definition, results unchanged",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let a continuous random process $X$ defined on $[0,1]$ be $(m+\\beta)$-smooth, $0\\le m, 0<\\beta\\le 1$, in quadratic mean for all $t>0$ and have an isolated singularity point at $t=0$. In addition, let $X$ be locally like a $m$-fold integrated $\\beta$-fractional Brownian motion for all non-singular points. We consider approximation of $X$ by piecewise Hermite interpolation splines with $n$ free knots (i.e., a sampling design, a mesh). The approximation performance is measured by mean errors (e.g., integrated or maximal quadratic mean errors). We construct a sequence of sampling designs with asymptotic approximation rate $n^{-(m+\\beta)}$ for the whole interval.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-05-19T09:00:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "62M86"
    ],
    "keywords": [
      "random process",
      "spline approximation",
      "maximal quadratic mean errors",
      "asymptotic approximation rate",
      "sampling design"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1004.5289A"
    }
  }
}