A formula for optimal integration in \(H^ 2\)
type: Item
instance of: scholarly article
title: A formula for optimal integration in \(H^ 2\)
zbMATH Open document ID: 0780.65012
DOI: 10.1016/0377-0427(93)90295-M
author: Jean-Paul Berrut
published in: Journal of Computational and Applied Mathematics
publication date: 1994-01-23T00:00:00Z
review text: Let \(H^ 2\) be the Hilbert space of the functions which are analytic within the unit disk and square integrable in the Lebesgue sense around the unit circle \(\partial D\), with the scalar product \((f,g) = (1/2\pi)\int_{\partial D}f(z)\overline{g(z)}dz\). For approximating the integral \(If:=\int_ \Gamma f(z)dz\) of a function \(f\in H^ 2\) along a path \(\Gamma \subset \mathbb{C}\), the author considers the integration formula (1) \(Qf = \sum^ n_{j=0}a_ jf_ j\), \(f_ j:= f(z_ j)\), where the \(z_ j\) are points on or outside \(\Gamma\), and \(a_ j \in \mathbb{C}\) are the weights. The author obtains an optimal integration formula of the type (1), i.e. he finds the weights \(a_ j\) for which the error functional \(Ef:= If-Qf\) has minimal norm on \(H^ 2\). He compares numerically the accuracy of the optimal formula with some well-known integration formulae. For points equidistant on a circle of radius \(r\) the formula allows the derivation of the barycentric formula. Numerical examples are considered.
Mathematics Subject Classification ID: 65D32
Mathematics Subject Classification ID: 65E05
Mathematics Subject Classification ID: 30E20
Mathematics Subject Classification ID: 30E10
zbMATH DE Number: 219120
zbMATH Keywords: optimal integration formula
zbMATH Keywords: Numerical examples
reviewed by: Dumitru Acu
MaRDI profile type: Publication
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full work available at URL: 0377-0427(93)90295-m
OpenAlex ID: W1966382077
description: scientific article; zbMATH DE number 219120