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New integral identities for orthogonal polynomials on the real line

Author: D. S. Lubinsky
Journal: Proc. Amer. Math. Soc. 139 (2011), 1743-1750
MSC (2010): Primary 42C05
Published electronically: October 18, 2010
MathSciNet review: 2763762
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Abstract: Let $\mu$ be a positive measure on the real line, with associated orthogonal polynomials $\left \{ p_{n}\right \}$ and leading coefficients $\left \{ \gamma _{n}\right \}$. Let $h\in L_{1}\left ( \mathbb {R}\right )$ . We prove that for $n\geq 1$ and all polynomials $P$ of degree $\leq 2n-2$, \begin{equation*} \int _{-\infty }^{\infty }\frac {P(t)}{p_{n}^{2}\left ( t\right ) } h\left ( \frac {p_{n-1}}{p_{n}} \left ( t\right ) \right ) dt=\frac {\gamma _{n-1}}{\gamma _{n}} \left ( \int _{-\infty }^{\infty }h\left ( t\right ) dt\right ) \left ( \int P\left ( t\right ) \text { }d\mu \left ( t\right ) \right ) . \end{equation*} As a consequence, we establish weak convergence of the measures on the left-hand side.

References [Enhancements On Off] (What's this?)

  • Louis de Branges, Hilbert spaces of entire functions, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1968. MR 0229011
  • René Carmona, One-dimensional Schrödinger operators with random or deterministic potentials: new spectral types, J. Funct. Anal. 51 (1983), no. 2, 229–258. MR 701057, DOI
  • G. Freud, Orthogonal Polynomials, Pergamon Press/Akademiai Kiado, Budapest, 1971.
  • I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London-Toronto, Ont., 1980. Corrected and enlarged edition edited by Alan Jeffrey; Incorporating the fourth edition edited by Yu. V. Geronimus [Yu. V. Geronimus] and M. Yu. Tseytlin [M. Yu. Tseĭtlin]; Translated from the Russian. MR 582453
  • Denis Krutikov and Christian Remling, Schrödinger operators with sparse potentials: asymptotics of the Fourier transform of the spectral measure, Comm. Math. Phys. 223 (2001), no. 3, 509–532. MR 1866165, DOI
  • D. S. Lubinsky, Universality limits for random matrices and de Branges spaces of entire functions, J. Funct. Anal. 256 (2009), no. 11, 3688–3729. MR 2514057, DOI
  • D. S. Lubinsky, Applications of New Geronimus Type Identities for Real Orthogonal Polynomials, Proc. Amer. Math. Soc. 138 (2010), 2125–2134.
  • Barry Simon, Orthogonal polynomials on the unit circle. Part 1, American Mathematical Society Colloquium Publications, vol. 54, American Mathematical Society, Providence, RI, 2005. Classical theory. MR 2105088
  • Barry Simon, Orthogonal polynomials with exponentially decaying recursion coefficients, Probability and mathematical physics, CRM Proc. Lecture Notes, vol. 42, Amer. Math. Soc., Providence, RI, 2007, pp. 453–463. MR 2352283, DOI

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Additional Information

D. S. Lubinsky
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160
MR Author ID: 116460
ORCID: 0000-0002-0473-4242

Keywords: Orthogonal polynomials on the real line, Geronimus type formula, Poisson integrals
Received by editor(s): March 23, 2010
Received by editor(s) in revised form: May 21, 2010
Published electronically: October 18, 2010
Additional Notes: This research was supported by NSF grant DMS1001182 and U.S.-Israel BSF grant 2008399
Communicated by: Walter Van Assche
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.