A note on asymptotic zero distribution of orthogonal polynomials.
type: Item
instance of: scholarly article
title: A note on asymptotic zero distribution of orthogonal polynomials.
zbMATH Open document ID: 1051.33007
DOI: 10.1016/S0024-3795(03)00657-8
author: William F. Trench
published in: Linear Algebra and its Applications
publication date: 2003-12-04T00:00:00Z
review text: Let \(a\in\mathbb{R}_+\) and \(b\in\mathbb{R}\) be constants, and consider a sequence of orthogonal polynomials given as \(q_{-1}(x)\equiv 0\), \(q_0(x)= 1\), and \[ aq_{n+1}(x)+ (b- x)q_n(x)+ aq_{n-1}(x)\equiv 0,\quad n\geq 1, \] with zeros \(y_{1n}< y_{2n}<\cdots< y_{nn}\) for \(q_n\). Together with the zeros the author considers the eigenvalues \(x_{1n}, x_{2n},\dots, x_{nn}\) of a symmetric \(n\times n\) matrix \((c_{ij})\) that satisfies (i) \(c_{ij}= 0\) unless \(| i- j|\) equals 0 or 1, and (ii) there are -- in a sense made precise in the paper -- not too many large differences \(| a- c_{i,i\pm 1}|\) and \(| b- c_{ii}|\). Then, the author's main result may be stated as follows \[ \sum^\infty_{r=1}| F(x_{rn})- F)y_{rn})|= o(n) \] for all real functions \(F\) that are bounded and continuous on some open interval \(I\supseteq [b- 2a, b+ 2a]\).
Mathematics Subject Classification ID: 33C52
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 42C05
zbMATH DE Number: 2013058
zbMATH Keywords: absolutely equally distributed
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: zero distribution
reviewed by: Per W. Karlsson
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2013058