{
  "id": "1701.04995",
  "version": "v1",
  "published": "2017-01-18T10:07:10.000Z",
  "updated": "2017-01-18T10:07:10.000Z",
  "title": "Christoffel formula for kernel polynomials on the unit circle",
  "authors": [
    "Cleonice F. Bracciali",
    "Andrei Martínez-Finkelshtein",
    "A. Sri Ranga",
    "Daniel O. Veronese"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "Given a nontrivial positive measure $\\mu$ on the unit circle, the associated Christoffel-Darboux kernels are $K_n(z, w;\\mu) = \\sum_{k=0}^{n}\\overline{\\varphi_{k}(w;\\mu)}\\,\\varphi_{k}(z;\\mu)$, $n \\geq 0$, where $\\varphi_{k}(\\cdot; \\mu)$ are the orthonormal polynomials with respect to the measure $\\mu$. Let the positive measure $\\nu$ on the unit circle be given by $d \\nu(z) = |G_{2m}(z)|\\, d \\mu(z)$, where $G_{2m}$ is a conjugate reciprocal polynomial of exact degree $2m$. We establish a determinantal formula expressing $\\{K_n(z,w;\\nu)\\}_{n \\geq 0}$ directly in terms of $\\{K_n(z,w;\\mu)\\}_{n \\geq 0}$. Furthermore, we consider the special case of $w=1$; it is known that appropriately normalized polynomials $K_n(z,1;\\mu) $ satisfy a recurrence relation whose coefficients are given in terms of two sets of real parameters $ \\{c_n(\\mu)\\}_{n=1}^{\\infty}$ and $ \\{g_{n}(\\mu)\\}_{n=1}^{\\infty}$, with $0<g_n<1 $ for $n\\geq 1$. The double sequence $\\{(c_n(\\mu), g_{n}(\\mu))\\}_{n=1}^{\\infty}$ characterizes the measure $\\mu$. A natural question about the relation between the parameters $c_n(\\mu)$, $g_n(\\mu)$, associated with $\\mu$, and the sequences $c_n(\\nu)$, $g_n(\\nu)$, corresponding to $\\nu$, is also addressed. Finally, examples are considered, such as the Geronimus weight (a measure supported on an arc of the unit circle), a class of measures given by basic hypergeometric functions, and a class of measures with hypergeometric orthogonal polynomials.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-01-18T10:07:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42C05",
      "33C47"
    ],
    "keywords": [
      "unit circle",
      "christoffel formula",
      "kernel polynomials",
      "basic hypergeometric functions",
      "conjugate reciprocal polynomial"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}