On the number of nodes in \(n\)-dimensional cubature formulae of degree 5 for integrals over the ball
type: Item
instance of: scholarly article
title: On the number of nodes in \(n\)-dimensional cubature formulae of degree 5 for integrals over the ball
zbMATH Open document ID: 1107.65025
author: Yanyan Li
published in: Journal of Computational and Applied Mathematics
publication date: 2004-08-16T00:00:00Z
review text: The paper deals with a problem associated with cubature formulas for the integral \(\int_B f(x)\,dx\), where \(B\) is the \(n\)-dimensional unit ball. Given the set of polynomials of degree \(m\) in \(n\) variables, it is an important question to find cubature formulas that are exact for all these polynomials. Particularly significant is the minimal formula, i.e.\ the formula with the smallest number of nodes. The knowledge about such minimal formulas is very limited at the moment. The paper under review gives some useful information about the minimal formulas in the case \(m=5\). The authors discuss classical lower bounds for the number of nodes of a minimal formula, show some cases when these bounds are sharp and improve the bounds in certain other cases, and they provide information about the location of the nodes in the cases when the classical bounds are attained. The proofs are based on methods using reproducing kernels and spherical 5-designs, and they provide additional useful information about spherical 5-designs.
reviewed by: Kai Diethelm
Mathematics Subject Classification ID: 65D32
Mathematics Subject Classification ID: 41A55
Mathematics Subject Classification ID: 41A63
zbMATH DE Number: 2091408
zbMATH Keywords: cubature
zbMATH Keywords: spherical designs
zbMATH Keywords: minimal formulas
MaRDI profile type: Publication
full work available at URL: j.cam.2003.12.024
OpenAlex ID: W2088807866
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DOI: 10.1016/J.CAM.2003.12.024
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description: scientific article; zbMATH DE number 2091408