Asymptotics of the Christoffel functions on a simplex in \(\mathbb{R}^d\)
type: Item
instance of: scholarly article
title: Asymptotics of the Christoffel functions on a simplex in \(\mathbb{R}^d\)
zbMATH Open document ID: 0943.42012
author: Yuan Xu
published in: Journal of Approximation Theory
publication date: 2000-09-03T00:00:00Z
review text: The author gives very advanced results on the Christoffel function \(\Lambda\) with respect to a weight \(W\), defined on a region \(\Omega\) or on the standard simplex \[ \Sigma^d=\{{\mathbf x}\in{\mathcal R}^d:x_1\geq 0,\ldots,x_d\geq 0,1-x_1-\cdots -x_d\geq 0\}. \] For a region \(\Omega\subset{\mathcal R}^d\), this function, given by \[ \Lambda_n(W;{\mathbf x})=\min_{P({\mathbf x})=1,P\in\Pi_n^d} \int_{\Omega} P^2({\mathbf y})W({\mathbf y}) d{\mathbf y}, \] plays an important role in the theory of orthogonal polynomials in several variables (it is the reciprocal of the reproducing kernel whose definition is independent of the choice of the orthonormal polynomial basis). First, in section 2, results are given for the Chebyshev weight \[ W_{\alpha}({\mathbf x})=w_{\alpha}x_1^{\alpha_1-1/2}\cdots x_d^{\alpha_d-1/2}(1-|{\mathbf x}|)^{\alpha_{d+1}-1/2},\;\alpha_i\geq 0, \] on the simplex \(\Sigma^d\) (the case \(\alpha_i=0\) only). In section 3 several very interesting results connecting various weights to \(W_0\) are derived. A nice paper.
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 1333085
zbMATH Keywords: orthogonal polynomials in several variables on a simplex
zbMATH Keywords: Christoffel function
zbMATH Keywords: asymptotics
reviewed by: Marcel G. de Bruin
MaRDI profile type: Publication
full work available at URL: jath.1998.3312
OpenAlex ID: W2016847492
cites work: Q4838051
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DOI: 10.1006/JATH.1998.3312
description: scientific article; zbMATH DE number 1333085