On the quadrature error in operational quadrature methods for convolutions
type: Item
instance of: scholarly article
title: On the quadrature error in operational quadrature methods for convolutions
zbMATH Open document ID: 0768.65090
DOI: 10.1007/BF01396219
author: Paul P. B. Eggermont
published in: Numerische Mathematik
publication date: 1993-02-08T00:00:00Z
full work available at URL: https://eudml.org/doc/133641
review text: An estimate is derived for the approximation of a Volterra integral operator witator with convolution like kernel by a discrete operator. Let \(T\) be the integral operator defined by \(Tx(t) = \int^ t_{-\infty} k(t-s)x(s)ds\), \(t\in\mathbb{R}\), and let \(T_ h\) be the discrete operator \(T_ hx(nh) = h\sum^ n_{m=0} k(n-m/h)x(mh)\), \(h > 0\). The Laplace transform \(K(s)\) of \(k(t)\) is assumed to be analytic in an obtuse sector \(\Sigma\) of the complex plane and (i) \(K(s) - K(0) = 0(| s|^ \alpha)\), \(| s| \to 0\), \(s\in \Sigma\), (ii) \(K(s) = 0(| s|^ \beta)\), \(| s| \to \infty\), \(s \in \Sigma\), \(\alpha,\beta > 0\). Let \(W^{p,M}(\mathbb{R})\) be the Sobolev space of functions \(x(t)\) with norm \(\| x\|_{W^{p,M}(\mathbb{R})} = \| x\|_{L^ p(\mathbb{R})} + \| x^{(M)}\|_{L^ p(\mathbb{R})}\), \(1 \leq p \leq \infty\). It is proved that \(T\), \(T_ h\) satisfy (iii) \(h^{1/p}\| T_ h x(nh) - Tx(nh)\|_{\ell^ p} \leq C p^ M\| x\|_{L^ p(M)}\) for \(x(t) \in W^{p,M}(\mathbb{R})\). When the weights \(k(n/h)\) are generated from \(K(s)\) by the method of \textit{C. Lubich} [Numer. Math. 52, 413-425 (1988; Zbl 0643.65094)]. An analogous result is proved for \(x(t)\) defined on \(R^ +\).
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 45D05
zbMATH DE Number: 118300
zbMATH Keywords: quadrature error
zbMATH Keywords: operational quadrature methods
zbMATH Keywords: Volterra integral operator
zbMATH Keywords: convolution like kernel
zbMATH Keywords: Laplace transform
reviewed by: John B. Butler jun.
MaRDI profile type: Publication
cites work: Q5799604
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description: scientific article; zbMATH DE number 118300