New integral identities for orthogonal polynomials on the real line

Lubinsky, D.

doi:10.1090/S0002-9939-2010-10601-9

New integral identities for orthogonal polynomials on the real line
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by D. S. Lubinsky PDF

Proc. Amer. Math. Soc. 139 (2011), 1743-1750 Request permission

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

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 10.1016/0022-1236(83)90027-7
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 10.1007/s002200100552
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 10.1016/j.jfa.2009.02.021
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, DOI 10.1090/coll054.1
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 10.1090/crmp/042/23