Reflection coefficients for the generalized Jacobi weight functions
type: Item
instance of: scholarly article
title: Reflection coefficients for the generalized Jacobi weight functions
zbMATH Open document ID: 0814.42013
author: Leonid Golinskii
published in: Journal of Approximation Theory
publication date: 1995-06-18T00:00:00Z
review text: Let \(w(t)= f(t) \prod^ N_ 1 | e^{it}- e^{it_ j}|^{\gamma_ j}\) be a generalized Jacobi weight on the unit circle, where \(\gamma_ j> -1\) and \(h\) is a smooth differentiable function. Consider the orthonormal polynomials \(\{\Phi_ n\}\) associated with \(w\) and the so-called reflection coefficients \(\Phi_ n(0)\) for them. These coefficients appear in the recurrence formula for the orthogonal polynomials and their behavior is very important in the theory. In the paper the very interesting estimate \[ 0< \limsup_{n\to \infty} n| \Phi_ n(0)|< \infty \] is proven. Then the result is transformed to the real line, where it reads that if \(\{a_ n\}\) and \(\{b_ n\}\) are the recursive coefficients corresponding to the orthogonal polynomials with respect to a generalized Jacobi weight, then \(| a_ n- 1/2|\) and \(| b_ n|\) are bounded by a constant times \(1/n\).
reviewed by: Vilmos Totik
Mathematics Subject Classification ID: 42C05
zbMATH DE Number: 636062
zbMATH Keywords: Jacobi weight
zbMATH Keywords: reflection coefficients
zbMATH Keywords: recurrence formula
zbMATH Keywords: orthogonal polynomials
MaRDI profile type: Publication
full work available at URL: jath.1994.1065
OpenAlex ID: W2082362630
DOI: 10.1006/JATH.1994.1065
description: scientific article; zbMATH DE number 636062