{
  "id": "1003.3336",
  "version": "v1",
  "published": "2010-03-17T09:51:11.000Z",
  "updated": "2010-03-17T09:51:11.000Z",
  "title": "A new approach to the asymptotics for Sobolev orthogonal polynomials",
  "authors": [
    "M. Alfaro",
    "J. J. Moreno-Balcazar",
    "A. Pena",
    "M. L. Rezola"
  ],
  "comment": "31 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "In this paper we deal with polynomials orthogonal with respect to an inner product involving derivatives, that is, a Sobolev inner product. Indeed, we consider Sobolev type polynomials which are orthogonal with respect to $$(f,g)=\\int fg d\\mu +\\sum_{i=0}^r M_i f^{(i)}(0) g^{(i)}(0), \\quad M_i \\ge 0,$$ where $\\mu$ is a certain probability measure with unbounded support. For these polynomials, we obtain the relative asymptotics with respect to orthogonal polynomials related to $\\mu$, Mehler--Heine type asymptotics and their consequences about the asymptotic behaviour of the zeros. To establish these results we use a new approach different from the methods used in the literature up to now. The development of this technique is highly motivated by the fact that the methods used when $\\mu$ is bounded do not work.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-03-17T09:51:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42C05",
      "33C45"
    ],
    "keywords": [
      "sobolev orthogonal polynomials",
      "sobolev inner product",
      "sobolev type polynomials",
      "mehler-heine type asymptotics",
      "asymptotic behaviour"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1003.3336A"
    }
  }
}