Integration with respect to a vector measure and function approximation
type: Item
instance of: scholarly article
title: Integration with respect to a vector measure and function approximation
zbMATH Open document ID: 1014.46012
DOI: 10.1155/S1085337501000227
published in: Abstract and Applied Analysis
publication date: 2002-08-13T00:00:00Z
full work available at URL: https://eudml.org/doc/50095
review text: Let \(\Sigma\) be the \(\sigma\)-algebra of Lebesgue measurable sets of \(\mathbb{R}\), \(\mu\) the Lebesgue measure. Let \((A_i)^n_1\) be a family of pairwise disjoint elements of \(\Sigma\) with \(0<\mu(A_i) <\infty\). The authors define a vector measure as \[ \lambda(A)= \sum^n_1\mu (A_j\cap A)\cdot e_i \] for \(A\in \Sigma\), where \((e_i)^n_1\) is the canonical basis of \(\mathbb{R}^n\). Let \(\Omega= \cup^n_1 A_i\) and \(f\in L^2(\Omega)\); one puts \[ \int_\Omega f d\lambda= \sum^n_1 \Bigl(\int_{A_i} f du\Bigr)e_i \] The authors show that this notion of an integral is useful for approximating elements of \(L^2(\Omega)\) by linear combination of an orthogonal system \((f_i)^n_1\).
reviewed by: Thanh Van Nguyen
Mathematics Subject Classification ID: 46G10
Mathematics Subject Classification ID: 46E99
zbMATH DE Number: 1780507
zbMATH Keywords: vector measures
zbMATH Keywords: function approximation
author: Lluís Miquel García Raffi
author: D. Ginestar
author: E. A. Sánchez-Pérez
MaRDI profile type: Publication
OpenAlex ID: W2095655899
description: scientific article; zbMATH DE number 1780507