The convergence of interpolatory product integration rules
type: Item
instance of: scholarly article
title: The convergence of interpolatory product integration rules
zbMATH Open document ID: 0629.41002
DOI: 10.1007/BF01939370
author: Philip Rabinowitz
published in: BIT
publication date: 1986-01-01T00:00:00Z
review text: The interpolatory product integration consists in approximating the integral \(I(kf)=\int^{1}_{-1}k(x)f(x)dx\) by a sequence of numerical integration rules of the form \(Q_ n^{(r,s)}f=\sum^{r+s}_{i=1- r}w_{in}(k)f(x_{in})\) where r,s\(\in \{0,1\}\) with \(- 1=x_{0n}<x_{1n}<...<x_{nn}<x_{n+1,n}=1\) and \(w_{in}(k)\) chosen such that \(Q_ n^{(r,s)}f=I(kf)\) if f is a polynomial of degree \(<n+r+s\). The author considers the convergence of \(Q_ n^{(r,s)}f\) to I(kf) for all \(f\in C[-1,1]\) in the case where the points \(x_{in}\) are the zeros of the orthogonal polynomials associated with a generalized smooth Jacobi weight w(x) and obtains the conditions on k and w(x) which insure the convergence.
Mathematics Subject Classification ID: 41A05
zbMATH DE Number: 4023929
zbMATH Keywords: interpolatory product integration
zbMATH Keywords: numerical integration rules
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: smooth Jacobi weight
MaRDI profile type: Publication
cites work: Q3323133
cites work: Product-integration rules and their convergence
cites work: The convergence of product integration rules
cites work: Mean Convergence of Lagrange Interpolation. III
cites work: On the numerical evaluation of singular integrals
cites work: Product-integration with the Clenshaw-Curtis and related points: Convergence properties
cites work: Properties of Interpolatory Product Integration Rules
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full work available at URL: https://doi.org/10.1007/bf01939370
OpenAlex ID: W2038426360
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description: scientific article; zbMATH DE number 4023929