{
  "id": "1001.0478",
  "version": "v1",
  "published": "2010-01-04T11:19:18.000Z",
  "updated": "2010-01-04T11:19:18.000Z",
  "title": "Orthogonal polynomials on several intervals: accumulation points of recurrence coefficients and of zeros",
  "authors": [
    "Franz Peherstorfer"
  ],
  "comment": "The last modifications and corrections of this manuscript were done by the author in the two months preceding this passing away in November 2009. The manuscript is not published elsewhere",
  "categories": [
    "math.CA",
    "math.CV"
  ],
  "abstract": "Let $E = \\cup_{j = 1}^l [a_{2j-1},a_{2j}],$ $a_1 < a_2 < ... < a_{2l},$ $l \\geq 2$ and set ${\\boldmath$\\omega$}(\\infty) =(\\omega_1(\\infty),...,\\omega_{l-1}(\\infty))$, where $\\omega_j(\\infty)$ is the harmonic measure of $[a_{2 j - 1}, a_{2 j}]$ at infinity. Let $\\mu$ be a measure which is on $E$ absolutely continuous and satisfies Szeg\\H{o}'s-condition and has at most a finite number of point measures outside $E$, and denote by $(P_n)$ and $({\\mathcal Q}_n)$ the orthonormal polynomials and their associated Weyl solutions with respect to $d\\mu$, satisfying the recurrence relation $\\sqrt{\\lambda_{2 + n}} y_{1 + n} = (x - \\alpha_{1 + n}) y_n -\\sqrt{\\lambda_{1 + n}} y_{-1 + n}$. We show that the recurrence coefficients have topologically the same convergence behavior as the sequence $(n {\\boldmath$\\omega$}(\\infty))_{n\\in \\mathbb N}$ modulo 1; More precisely, putting $({\\boldmath$\\alpha$}^{l-1}_{1 + n}, {\\boldmath$\\lambda$}^{l-1}_{2 + n}) = $ $(\\alpha_{[\\frac{l 1}{2}]+1+n},...,$ $\\alpha_{1+n},...,$ $\\alpha_{-[\\frac{l-2}{2}]+1+n},$ $\\lambda_{[\\frac{l-2}{2}]+2+n},$ $...,\\lambda_{2+n},$ $...,$ $\\lambda_{-[\\frac{l-1}{2}]+2+n})$ we prove that $({\\boldmath$\\alpha$}^{l-1}_{1 + n_\\nu},$ ${\\boldmath$\\lambda$}^{l-1}_{2 + n_\\nu})_{\\nu \\in \\mathbb N}$ converges if and only if $(n_\\nu {\\boldmath$\\omega$}(\\infty))_{\\nu \\in \\mathbb N}$ converges modulo 1 and we give an explicit homeomorphism between the sets of accumulation points of $({\\boldmath$\\alpha$}^{l-1}_{1 + n}, {\\boldmath$\\lambda$}^{l-1}_{2 + n})$ and $(n{\\boldmath$\\omega$}(\\infty))$ modulo 1.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-01-04T11:19:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "accumulation points",
      "recurrence coefficients",
      "orthogonal polynomials",
      "point measures outside",
      "finite number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1001.0478P"
    }
  }
}