On the convergence of Filon quadrature
type: Item
instance of: scholarly article
title: On the convergence of Filon quadrature
zbMATH Open document ID: 1193.41032
author: B. E. Eshmatov
published in: Journal of Computational and Applied Mathematics
publication date: 2010-06-10T00:00:00Z
review text: Given \(k\in \mathbb{R}\), it is assumed that the integral \(\int_{-1}^1 e^{\mathbf{i}kg(x)}\pi(x)dx\) can be evaluated for every polynomial \(\pi\). Therefore, a suitable polynomial quadrature rule can be applied to calculate \(Q(f)=\int_{-1}^1 e^{\mathbf{i}kg(x)}f(x)dx\) when \(f\) is holomorphic on \(\mathcal{G}_f\) and \(g\) is holomorphic on \(\mathcal{G}_f\), where \(\mathcal{G}_g\) and \(\mathcal{G}_f\) are neighborhoods of \([-1,1]\), with \(\mathcal{G}_g \subset \mathcal{G}_f\). Let \(\triangle=\{-1,\dots,-1,1,\dots,1\}\) and \(I_\triangle\) be the interpolation operator of Hermite type associated with the endpoints \(\pm 1\), considering that each of the nodes \(\pm 1\) appears \(p\) times. The pure Filon quadrature (PFQ) of order \(n=2p\) is that given by \(Q(I_\triangle(f))\). The paper is concerned with the convergence of this type of quadrature when \(g'(x)\neq 0\) for all \(x\in[-1,1]\), \(g\) and \(1/g'\) are holomorphic on \(\mathcal{G}_g\), and \(1/g'\in L^\infty(\mathcal{G}_g)\). Assuming an additional condition on \(\mathcal{G}_f\) the PFQ is convergent as \(p\to \infty\). The article shows the advantage of clustering interpolation points near the endpoints \(\pm 1\), so the PFQ represents the extreme case. It is proved that the size of \(\mathcal{G}_f\) plays a role in obtaining convergence of the PFQ as \(n\to \infty\). To create convergent Filon-type quadrature for integrands \(f\) with small domain of analyticity, the author presents two different ways of doing it. One is by considering a composite quadrature, and the other by inserting additional nodes in \([-1,1]\). The numerical stability of the Filon quadrature is not analyzed, however, some examples are given to illustrate the convergence properties.
Mathematics Subject Classification ID: 41A55
Mathematics Subject Classification ID: 65D32
zbMATH DE Number: 5719641
zbMATH Keywords: highly oscillatory integrands
zbMATH Keywords: filon quadrature
reviewed by: J. Illán
MaRDI profile type: Publication
full work available at URL: j.cam.2009.08.017
OpenAlex ID: W2154860675
cites work: On quadrature methods for highly oscillatory integrals and their implementation
cites work: Efficient quadrature of highly oscillatory integrals using derivatives
cites work: Q5291957
cites work: The Construction of cubature rules for multivariate highly oscillatory integrals
cites work: Q3848301
DOI: 10.1016/J.CAM.2009.08.017
description: scientific article; zbMATH DE number 5719641