What distributions of points are possible for convergent sequences of interpolatory integration rules!
type: Item
instance of: scholarly article
title: What distributions of points are possible for convergent sequences of interpolatory integration rules!
zbMATH Open document ID: 0770.41001
DOI: 10.1007/BF01229335
published in: Constructive Approximation
publication date: 1993-04-01T00:00:00Z
review text: Suppose that for \(n\geq 1\), \(I_ n[f]=\sum^ n_{s=1}\omega_{jn}f(x_{jn})\) is an interpolatory quadrature formula so that \(I_ n[P]=\int^ 1_{-1}P(x)dx\) for all \(P\in\pi_{n-1}\). Furthermore, if we suppose that \(\lim_{n\to\infty}In[f]=\int^ 1_{- 1}f(x)dx\) for each \(f\in C[-1,1]\), then in all the classical examples, the nodes \(\{x_{jn}\}\) are found to have arc sine distribution. The authors show that this is not the general case. They show that the set of all possible limit distributions has the form \({1\over 2}[v(x)dx+dv(x)]\) where \(\nu\) is an arbitrary probability measure on \([-1,1]\) and \(v(x)dx={dx\over\pi\sqrt{1-x^ 2}}\). To formulate their main theorem, they denote by \([-1,1]\) the set of all nonnegative Borel measure \(\mu\) on \([-1,1]\) with \(\mu[-1,1]=1\). \(\mu\) is called a probability measure and is also denoted by \(d\mu\). Given a sequence \(\{\nu_ n\}^ \infty_ 1\subset{\mathcal M}[-1,1]\) and \(\nu\in{\mathcal M}[-1,1]\), \(d\nu\) is called a weak limit of a subsequence of \(\{d\nu_ n\}^ \infty_ 1\) iff there exists an infinite subsequence of positive integers such that \(d\nu_ n@>*>>d\nu\) as \(n\to\infty\), \(n\in{\mathcal N}\). In other words \(\forall f\in C[-1,1]\) \[ \lim_{n\to\infty,m\in{\mathcal N}}\int^ 1_{- 1}f(x)d_{\nu_ n}(x)=\int^ 1_{-1}f(x)d\nu(x). \] The main result of the authors is: Theorem 1. (1) Let \(\{I_ n\}^ \infty_ 1\) be a sequence of interpolatory integration rules with \(\lim_{n\to\infty}\| I_ n\|^{1/n}=1\), \(\| I_ n\|=\sum^ n_{j=1}|\omega_{jn}|\), then every weak limit of a subsequence of the corresponding measures \(\{d\mu_ n\}^ \infty_ 1\) has the form \({1\over 2}[d\nu(x)+v(x)dx]\) for some \(\nu\in M[-1,1]\). (2) Conversely, given any \(\nu\in{\mathcal M}[-1,1]\), there exists a sequence \(\{I_ n\}_ 1^ \infty\) of interpolatory quadrature rules with all positive weights, (i.e., \(\omega_{jn}\geq 0\), \(1\leq j\leq n)\) such that the corresponding measures \(\{d\mu_ n\}^ \infty_ 1\) satisfy \[ d\mu_ n@>*>>…
reviewed by: A. Sharma
Mathematics Subject Classification ID: 41A05
Mathematics Subject Classification ID: 41A55
zbMATH DE Number: 146901
zbMATH Keywords: distribution of points
zbMATH Keywords: potential theory
zbMATH Keywords: interpolatory quadrature formula
zbMATH Keywords: weak limit
zbMATH Keywords: quadrature rules
author: Thomas Bloom
author: Doron S. Lubinsky
author: Herbert Stahl
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01229335
OpenAlex ID: W2045465202
description: scientific article; zbMATH DE number 146901