On Christoffel type functions for \(L_m\) extremal polynomials. I
type: Item
instance of: scholarly article
title: On Christoffel type functions for \(L_m\) extremal polynomials. I
zbMATH Open document ID: 0947.41021
author: Yingguang Shi
published in: Journal of Approximation Theory
publication date: 2000-11-05T00:00:00Z
review text: Let \(m\in \mathbb{N}\) be even and \(M:=\{ 0,2,4,\dots{},m-2\} .\) For \(n\in \mathbb{N}\) let \(\mathbf{P}_{n}\) be the set of all polynomials of degree \(\leq n\) and \(\mathbf{P}_{n}^{\ast }\) the subset of \(\mathbf{P}_{n}\) formed by polynomials having only real zeros. Let \(x\in \mathbb{R}\) and \(\mathbf{P}_{n}^{\ast }( x) :=\{ P\in \mathbf{P}_{n}^{\ast }:P( x) =1\} .\) For \(j\in M\) denote \[ A_{j}( P,x;t) :=A_{jnm}( P,x;t) :=\frac{1}{j!}( t-x) ^{j}B_{j}( P,x;t) ( P( t)) ^{m}, \] where \(P\in \mathbf{P}_{n-1},\) \(P( x) =1,\) \(B_{j}( P,x;\cdot) \in \mathbf{P}_{m-j-2}\) and \(A_{j}^{( i) }( P,x;x) =\delta _{ij},\) \(i=0,1,2,\dots{},m-2.\) The function defined by \(\lambda _{jnm}( d\alpha ,x) =\inf_{P\in \mathbf{P}_{n-1}^{\ast }( x) }\int_{\mathbb{R} }A_{j}( P;x;t) d\alpha ( t) ,\) \(x\in \mathbb{R},\) where \(\alpha :\mathbb{R}\rightarrow \mathbb{R}\) is a nondecreasing function with infinitely many points of increase such that all moments of \(d\alpha\) are finite is called Christoffel type function with respect to \(d\alpha \). The author proves that the preceeding infinum is attained for a unique polynomial \(P\in \mathbf{P}_{n-1}^{\ast }( x) \) with effective degree \(\geq n-2,\) and presents some properties of the functions \(\lambda_{jnm}( d\alpha ,\cdot) ,\) \(j\in M.\)
Mathematics Subject Classification ID: 41A50
zbMATH DE Number: 1443206
zbMATH Keywords: Christoffel type functions
reviewed by: Costică Mustăţa
MaRDI profile type: Publication
full work available at URL: jath.1999.3427
OpenAlex ID: W1964539254
cites work: Generalized Gaussian quadrature formulas
cites work: Géza Freud, orthogonal polynomials and Christoffel functions. A case study
cites work: On the fine and rough theory of Lagrange type interpolation of higher order
cites work: Q5595759
DOI: 10.1006/JATH.1999.3427
description: scientific article; zbMATH DE number 1443206