A note on the optimal quadrature in \(H^ p\)
type: Item
instance of: scholarly article
title: A note on the optimal quadrature in \(H^ p\)
zbMATH Open document ID: 0546.41026
DOI: 10.1007/BF01410113
author: Jan-Erik Andersson
author: Borislav D. Bojanov
published in: Numerische Mathematik
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/132937
review text: In the paper we study quadrature formulae of the form \(\int^{1}_{- 1}f(x)dx\approx\sum^{n}_{k=1}\sum^{\nu_ k-1}_{\lambda =0}a_{k\lambda}f^{(\lambda)}(x_ k)\) where \(-1<x_ 1<...<x_ n<1\). The existence of optimal nodes with preassigned multiplicities is proved for the Hardy spaces \(H^ p\) with \(1<p<\infty\). This result is then used to show that the exact order of convergence for the optimal quadrature formula with N nodes (including multiplicity) is \(N^{1/(2q)}\exp (- \pi\sqrt{N/q})\) where \(1/p+1/q=1\) and 1\(\leq p\leq\infty \).
Mathematics Subject Classification ID: 41A55
zbMATH DE Number: 3869922
zbMATH Keywords: quadrature formulae
zbMATH Keywords: optimal nodes
zbMATH Keywords: Hardy spaces
MaRDI profile type: Publication
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OpenAlex ID: W1986827392
description: scientific article; zbMATH DE number 3869922