On the convergence of interpolatory-type quadrature rules for evaluating Cauchy integrals
type: Item
instance of: scholarly article
title: On the convergence of interpolatory-type quadrature rules for evaluating Cauchy integrals
zbMATH Open document ID: 1024.65022
DOI: 10.1016/S0377-0427(02)00481-8
author: B. I. Yun
author: Philsu Kim
published in: Journal of Computational and Applied Mathematics
publication date: 2003-02-23T00:00:00Z
review text: The Cauchy principal value integral \(Q(f; t)\) with respect to the singularity \(t\) is transformed to \(H(h;x)\), where \(h(y)= f(\cos y)\) and \(t= \cos x\), and approximated by quadrature rules \(H_N(h;x)\) based on trigonometric interpolations. The \(N+1\) weights depend also from the singularity and the size of the stability factor \(\Lambda_N(x)= \sum''|\omega^N_k(x)|\) is of great importance for stability and convergence. The main results are the almost optimal estimate of the stability factor and the estimate of the remainder term \[ \Lambda_N(x)\leq C\Biggl({\log N+1\over \sin^2x}\Biggr)\quad\text{and}\quad |R_N(h; x)|\leq CM{N\log N\over 2^{N-1}}, \] respectively. Further, if the derivative \(f^{(p)}(t)\) satisfies the Hölder continuity of order \(\rho\), then the remainder term satisfies \(O((A+B\log N+ N^{2\nu})/N^{p+\rho})\).
reviewed by: Rudolf Scherer
Mathematics Subject Classification ID: 65D32
Mathematics Subject Classification ID: 41A55
Mathematics Subject Classification ID: 41A80
zbMATH DE Number: 1872924
zbMATH Keywords: Cauchy principal value integral
zbMATH Keywords: quadrature rule
zbMATH Keywords: trigonometric interpolation
zbMATH Keywords: Lebesgue constant
zbMATH Keywords: stability
zbMATH Keywords: convergence
zbMATH Keywords: remainder estimates
MaRDI profile type: Publication
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full work available at URL: s0377-0427(02)00481-8
OpenAlex ID: W2084281370
description: scientific article; zbMATH DE number 1872924