Eigenvalues of Toeplitz matrices associated with orthogonal polynomials
type: Item
instance of: scholarly article
title: Eigenvalues of Toeplitz matrices associated with orthogonal polynomials
zbMATH Open document ID: 0634.33017
DOI: 10.1016/0021-9045(87)90044-X
author: Walter Van Assche
published in: Journal of Approximation Theory
publication date: 1987-01-01T00:00:00Z
review text: In this significant paper, the author extends earlier results of Grenander, Szegö and Nevai relating zero distribution of orthogonal polynomials to the distribution of eigenvalues of related Toeplitz matrices. His results treat simultaneously orthogonal polynomials on finite and infinite intervals. Let \[ p_ n(x)=\gamma_ n\prod^{n}_{j=1}(x-x_{jn}),\quad \gamma_ n>0, \] denote the nth orthonormal polynomial for a positive measure \(d\alpha\) (x) on \({\mathbb{R}}\), \(n=0,1,2,..\).. Let g be a real valued function such that for \(k=0,1,2,...,x^ kg(x)\in L_ 1(d\alpha)\). Let \(\{c_ n\}^{\infty}_{n=1}\) be a sequence of positive numbers and introduce the \(n\times n\) Toeplitz matrix \[ T_ n(g)=[\int^{\infty}_{- \infty}p_ j(x)p_ k(x)g(x/c_ n)d\alpha (x)]^{n-1}_{j,k=0}, \] with eigenvalues \(\{\Lambda_{jn}\}^ n_{j=1}\), repeated according to multiplicity. Define probability measures \(\mu_ n\) and \(\nu_ n\) placing mass 1/n at \(x_{ij}/c_ n\) and \(\Lambda_{jn}\) respectively: So \[ \mu_ n=n^{-1}\sum^{n}_{j=1}\delta_{x_{jn}/c_ n};\quad \nu_ n=n^{-1}\sum^{n}_{j=1}\delta_{\Lambda_{jn}}, \] (\(\delta=\) Dirac delta). The main theorem states that if \(\mu_ n\) converges weakly to \(\mu\), and \(c_ n^{-1}(\gamma_{n-1}/\gamma_ n)\) is \(o(n^{1/2})\), and g is bounded with discontinuities in a set of \(\mu\)- measure zero, then \(\nu_ n\) converges weakly to a probability measure \(\nu\). More precisely, for every f continuous on \({\mathbb{R}}\), \[ \lim_{n\to \infty}n^{- 1}\sum^{n}_{j=1}f(\Lambda_{jn})=\int^{\infty}_{- \infty}f(g(x))d\mu (x). \]
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 41A55
Mathematics Subject Classification ID: 15B48
zbMATH DE Number: 4032104
zbMATH Keywords: eigenvalue distribution
zbMATH Keywords: zero distribution
zbMATH Keywords: eigenvalues
zbMATH Keywords: Toeplitz matrices
zbMATH Keywords: Toeplitz matrix
reviewed by: Doron S. Lubinsky
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4032104