Eigenvalue integrodifferential equations for orthogonal polynomials on the real line
نویسندگان
چکیده
منابع مشابه
Eigenvalue Integro-Differential Equations for Orthogonal Polynomials on the Real Line∗
The one-dimensional harmonic oscillator wave functions are solutions to a SturmLiouville problem posed on the whole real line. This problem generates the Hermite polynomials. However, no other set of orthogonal polynomials can be obtained from a Sturm-Liouville problem on the whole real line. In this paper we show how to characterize an arbitrary set of polynomials orthogonal on (−∞,∞) in terms...
متن کاملNew Integral Identities for Orthogonal Polynomials on the Real Line
Let be a positive measure on the real line, with associated orthogonal polynomials fpng and leading coe¢ cients f ng. Let h 2 L1 (R) . We prove that for n 1 and all polynomials P of degree 2n 2, Z 1 1 P (t) pn (t) h pn 1 pn (t) dt = n 1 n Z 1 1 h (t) dt Z P (t) d (t) : As a consequence, we establish weak convergence of the measures in the lefthand side. Orthogonal Polynomials on the real line, ...
متن کاملAsymptotics of Derivatives of Orthogonal Polynomials on the Real Line
We show that uniform asymptotics of orthogonal polynomials on the real line imply uniform asymptotics for all their derivatives. This is more technically challenging than the corresponding problem on the unit circle. We also examine asymptotics in the L2 norm. 1. Results Let μ be a nite positive Borel measure on [−1, 1] and let {pn}n=0 denote the corresponding orthonormal polynomials, so that ∫...
متن کاملZeros of orthogonal polynomials on the real line
Let pnðxÞ be the orthonormal polynomials associated to a measure dm of compact support in R: If EesuppðdmÞ; we show there is a d40 so that for all n; either pn or pnþ1 has no zeros in ðE d;E þ dÞ: If E is an isolated point of suppðmÞ; we show there is a d so that for all n; either pn or pnþ1 has at most one zero in ðE d;E þ dÞ:We provide an example where the zeros of pn are dense in a gap of su...
متن کامل2 00 2 A connection between orthogonal polynomials on the unit circle and matrix orthogonal polynomials on the real line
Szeg˝ o's procedure to connect orthogonal polynomials on the unit circle and orthogonal polynomials on [−1, 1] is generalized to nonsymmetric measures. It generates the so-called semi-orthogonal functions on the linear space of Laurent polynomials Λ, and leads to a new orthogonality structure in the module Λ × Λ. This structure can be interpreted in terms of a 2 × 2 matrix measure on [−1, 1], a...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Journal of Mathematical Physics
سال: 1995
ISSN: 0022-2488,1089-7658
DOI: 10.1063/1.531016