New integral identities for orthogonal polynomials on the real line
نویسندگان
چکیده
منابع مشابه
New 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, ...
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Let be a positive measure on the real line, with associated orthogonal polynomials fpng. Let Im a 6= 0. Then there is an explicit constant cn such that for all polynomials P of degree at most 2n 2, cn Z 1 1 P (t) jpn (a) pn 1 (t) pn 1 (a) pn (t)j dt = Z P d : In this paper, we provide a self-contained proof of the identity. Moreover, we apply the formula to deduce a weak convergence result, a d...
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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 ∫...
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ژورنال
عنوان ژورنال: Proceedings of the American Mathematical Society
سال: 2011
ISSN: 0002-9939
DOI: 10.1090/s0002-9939-2010-10601-9