{
  "id": "0909.0619",
  "version": "v1",
  "published": "2009-09-03T10:30:00.000Z",
  "updated": "2009-09-03T10:30:00.000Z",
  "title": "Orthogonal polynomials associated with an inverse quadratic spectral transform",
  "authors": [
    "M. Alfaro",
    "F. Marcellan",
    "A. Pena",
    "M. L. Rezola"
  ],
  "comment": "21 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "Let $\\{P_n \\}_{n\\ge0}$ be a sequence of monic orthogonal polynomials with respect to a quasi--definite linear functional $u$ and $\\{Q_n \\}_{n\\ge0}$ a sequence of polynomials defined by $$Q_n(x)=P_n(x)+s_n P_{n-1}(x)+t_n P_{n-2}(x),\\quad n\\ge1,$$ with $t_n \\not= 0$ for $n\\ge2$. We obtain a new characterization of the orthogonality of the sequence $\\{Q_n \\}_{n\\ge0}$ with respect to a linear functional $v$, in terms of the coefficients of a quadratic polynomial $h$ such that $h(x)v= u$. We also study some cases in which the parameters $s_n$ and $t_n$ can be computed more easily, and give several examples. Finally, the interpretation of such a perturbation in terms of the Jacobi matrices associated with $\\{P_n \\}_{n\\ge0}$ and $\\{Q_n \\}_{n\\ge0}$ is presented.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-09-03T10:30:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42C05",
      "33C45"
    ],
    "keywords": [
      "inverse quadratic spectral transform",
      "monic orthogonal polynomials",
      "quasi-definite linear functional",
      "jacobi matrices"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0909.0619A"
    }
  }
}