Rational moment problems for compact sets
type: Item
instance of: scholarly article
title: Rational moment problems for compact sets
zbMATH Open document ID: 0812.41016
author: James D. Chandler
published in: Journal of Approximation Theory
publication date: 1994-09-27T00:00:00Z
review text: The author treats the following `rational' moment problem: given distinct real numbers \(\lambda_ 1,\dots,\lambda_ p\) and real numbers \(c_ 0\), \(c_ j^{(i)}\) \((j\geq 1,\;1\leq i\leq p)\) and a non- empty compact subset \({\mathbf K}\) of the real line, find a non-negative Borel measure \(\mu\) supported on \({\mathbf K}\) with \[ c_ 0= \int_{{\mathbf K}} d\mu(t),\;c_ j^{(i)}= \int_{{\mathbf K}}(t- \lambda_ i)^{-j} d\mu(t). \] By looking at the truncated moment problem (restrict \(j: 1\leq j\leq 2n_ i\), \(2n_ i\) positive even integers), the author derives necessary and sufficient conditions for the solvability of the rational moment problem in terms of the positivity of certain linear functionals on the space of rational functions of the form \[ P(t)/\prod_{i=1}^ p (t-\lambda_ i)^{2n_ i} \] with \(P(t)\) a polynomial of degree \(\leq 2(n_ 1+\cdots+ n_ p)\). Also the case with countably many \(\lambda\)'s is treated. Moreover the finite and countably infinite case of the problem with a pole at infinity (replace \(c_ 0\), by \(c_ j^{(0)}\) and the conditions for those moments are \(c_ j^{(0)}=\int_{{\mathbf K}} t^ j d\mu(t)\)) are solved. The paper concludes with the construction of a positive Borel measure \(\mu\) with finite total mass and support equal to the closed interval \([0,2]\) such that \((t- \lambda_ i)^{-j}\), \(t^ j\) \((i\geq 0, j\geq 0)\) belong to \(L^ 1(\mu)\), where the \(\lambda\)'s satisfy \(0< \lambda_ 1<\lambda_ 2<\cdots< \lambda_ 0< 1\) and \(\lambda_ n\to \infty\) as \(n\to \infty\). A nicely written, interesting paper.
Mathematics Subject Classification ID: 41A20
zbMATH DE Number: 645058
zbMATH Keywords: non-negative polynomials
zbMATH Keywords: non-negative linear functionals
zbMATH Keywords: rational moment problem
reviewed by: Marcel G. de Bruin
MaRDI profile type: Publication
full work available at URL: jath.1994.1114
OpenAlex ID: W2002838901
DOI: 10.1006/JATH.1994.1114
description: scientific article; zbMATH DE number 645058