A shortcut to asymptotics for orthogonal polynomials
type: Item
instance of: scholarly article
title: A shortcut to asymptotics for orthogonal polynomials
zbMATH Open document ID: 0804.33008
DOI: 10.1016/0377-0427(94)90301-8
author: Thomas Dehn
published in: Journal of Computational and Applied Mathematics
publication date: 1994-09-22T00:00:00Z
review text: Let \(q_ k\) be the sequence of orthonormal polynomials with respect to the positive measure \(\mu\) with real support \(E\). These satisfy the well known three term recurrence relation \(zq_ n(z)= \beta_ n q_{n+1} (z)+ \alpha_ n q_ n(z)+ \beta_{n-1} q_{n-1}(z)\). The Nevai class \(M(0,1)\) is the class of measures for which \(\alpha_ n\) tends to 0 and \(\beta_ n\) to \(1\over 2\). Nevai (1979) showed that for \(\mu\in M(0,1)\), the support \(E\) consists of \([-1,1]\) and at most a denumerable set of points outside this interval. Moreover, if \(\Phi\) is the conformal map of \(\mathbb{C}\setminus [-1,1]\) onto \(| z|>1\), then \(q_{n+1} (z)/ q_ n(z)\) tends to \(\Phi(z)\) locally uniformly in \(\mathbb{C} \setminus E\) and to \(1/ \Phi(z)\) pointwise in \(E\setminus [-1,1]\). In this paper new proofs for these results and related ones are provided. They rely on writing the recurrence relation as a 2 by 2 system of first order recurrences. The eigenvalues of the limit of the recurrence matrix are precisely \(\Phi(z)\) and \(1/\Phi(z)\). The ratio asymptotics thus correspond to the convergence to eigenvalues and this is related to general theorems for recurrence relations, the power method for eigenvalue computation and Poincaré's theorem. An historical survey about these topics and preliminary results about the zero distribution and the normal family properties of the ratio's are also included. In the last section, convergence results for diagonal Padé approximants to Hamburger functions are derived. The ratio asymptotics allow to accelerate this sequence of approximants by the \(\Delta^ 2\) method. This is also proved.
reviewed by: Adhemar Bultheel
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 42C05
zbMATH DE Number: 643755
zbMATH Keywords: difference equations
zbMATH Keywords: recurrence relation
zbMATH Keywords: ratio asymptotics
zbMATH Keywords: Poincaré's theorem
zbMATH Keywords: Padé approximants
zbMATH Keywords: \(\Delta^ 2\) method
MaRDI profile type: Publication
full work available at URL: 0377-0427(94)90301-8
OpenAlex ID: W2088073174
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description: scientific article; zbMATH DE number 643755