Ratio asymptotics and quadrature formulas
type: Item
instance of: scholarly article
title: Ratio asymptotics and quadrature formulas
zbMATH Open document ID: 0896.42017
DOI: 10.1007/BF02678469
published in: Constructive Approximation
publication date: 1998-09-27T00:00:00Z
review text: Suppose \(p_n\) \((n=0,1,2,\ldots)\) is a sequence of orthogonal polynomials on the real line, satisfying a three-term recurrence relation \(tp_n(t) = a_{n+1}p_{n+1}(t)+b_np_n(t)+a_np_{n-1}(t)\). The author gives a method for obtaining the asymptotic behaviour of the ratio \(s_n(z)/p_n(z)\) for a comparison sequence \(s_n\) \((n=0,1,2,\ldots)\) of polynomials. Crucial is the investigation of modified moments \(\int r_m(t) d\mu(t)\) where the \(r_m\) are polynomials suitably chosen in terms of the given polynomials \(s_n\) and the orthogonal polynomials \(p_n\) which one wants to study. The method gives ratio asymptotics for a large class of orthogonal polynomials in the class \(M\) of converging recurrence coefficients, but also for unbounded recurrence coefficients (even with exponential growth). In an appendix the author also shows how the technique can be used when the Fourier coefficients of \(s_n(t) = \sum_{k=0}^n c_k p_k(t)\) behave nicely, which in particular gives ratio asymptotics for compact perturbations of orthogonal polynomials.
Mathematics Subject Classification ID: 42C05
zbMATH DE Number: 1023015
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: quadrature
zbMATH Keywords: ratio asymptotics
author: A. J. Durán
reviewed by: Walter Van Assche
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1023015