{
  "id": "math/0606589",
  "version": "v2",
  "published": "2006-06-23T10:24:10.000Z",
  "updated": "2007-05-02T14:50:56.000Z",
  "title": "Sobolev orthogonal polynomials: balance and asymptotics",
  "authors": [
    "M. Alfaro",
    "J. J. Moreno-Balcazar",
    "A. Pena",
    "M. L. Rezola"
  ],
  "comment": "20 pages. Changed content",
  "categories": [
    "math.CA"
  ],
  "abstract": "Let $\\mu_0$ and $\\mu_1$ be measures supported on an unbounded interval and $S_{n,\\lambda_n}$ the extremal varying Sobolev polynomial which minimizes \\begin{equation*} < P, P >_{\\lambda_n}=\\int P^2 d\\mu_0 + \\lambda_n \\int P'^2 d\\mu_1, \\quad \\lambda_n >0 \\end{equation*} \\noindent in the class of all monic polynomials of degree $n$. The goal of this paper is twofold. On one hand, we discuss how to balance both terms of this inner product, that is, how to choose a sequence $(\\lambda_n)$ such that both measures $\\mu_0$ and $\\mu_1$ play a role in the asymptotics of $(S_{n, \\lambda_n}).$ On the other, we apply such ideas to the case when both $\\mu_0$ and $\\mu_1$ are Freud weights. Asymptotics for the corresponding $S_{n, \\lambda_n}$ are computed, illustrating the accuracy of the choice of $\\lambda_n .$",
  "revisions": [
    {
      "version": "v2",
      "updated": "2007-05-02T14:50:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42C05"
    ],
    "keywords": [
      "sobolev orthogonal polynomials",
      "asymptotics",
      "extremal varying sobolev polynomial",
      "inner product",
      "monic polynomials"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......6589A"
    }
  }
}