Distribution of nodes on algebraic curves in \({\mathbb C}^N\).
type: Item
instance of: scholarly article
title: Distribution of nodes on algebraic curves in \({\mathbb C}^N\).
zbMATH Open document ID: 1044.32026
publication date: 2003-12-08T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/116075
review text: Let \(A\subset\mathbb C^N\) be an irreducible curve and let \(K\subset A\) be a non-polar compact set. For \(d=0,1,\dots\), let \(m_d\) be the dimension of the space \(P_d| A\) of all complex polynomials of degree at most \(d\) restricted to \(A\). It is known that for sufficiently large \(d\) we have \(m_d=dD+c\), where \(D\) is the degree of \(A\) and \(c\) is an integer. Consider Lagrange interpolation polynomials \(L_df(z)= \sum_{j=1}^{m_d}f(A_{d,j})\ell^{(d)}_j(z)\) with nodes \(A_{d,j}\in K\). Let \(\varLambda_d:=\| \sum_{j=1}^{m_d}| \ell^{(d)}_j| \| _K\). Assume that the nodes are chosen in such a way that \(\limsup_{d\to+\infty}\varLambda_d^{1/d}\leq1\). Then \((1/m_d)\sum_{j=1}^{m_d}\delta_{A_{d,j}}\overset{\text{weak--}\ast}\)
Mathematics Subject Classification ID: 32U05
Mathematics Subject Classification ID: 31C10
Mathematics Subject Classification ID: 41A05
zbMATH DE Number: 2014680
zbMATH Keywords: algebraic curve
zbMATH Keywords: Lebesgue constant
author: Thomas Bloom
author: Norm Levenberg
reviewed by: Marek Jarnicki
MaRDI profile type: Publication
OpenAlex ID: W2599733726
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DOI: 10.5802/AIF.1982
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published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 2014680