Optimal cubature formulas on compact homogeneous manifolds
type: Item
instance of: scholarly article
title: Optimal cubature formulas on compact homogeneous manifolds
zbMATH Open document ID: 1185.41026
published in: Journal of Functional Analysis
publication date: 2009-09-10T00:00:00Z
review text: Let \(\mathbb{M}^d\) be a~two-point compact homogeneous manifold in \(\mathbb{R}^{d+1}\) and let \(W_{\infty}^r(\mathbb{M}^d)\) stands for the class of \(r\)-times differentiable functions \(f:\mathbb{M}^d\to\mathbb{R}\) with \(\|f\|_{\infty}\leq 1\). The main result states that for any \(r>0\) and \(\varepsilon>0\) \[ \inf_{\{x_1,\dots,x_n\}\subset\mathbb{M}^d}\inf_{\{\alpha_1,\dots,\alpha_n\}\subset\mathbb{R}}\;\sup_{f\in W_{\infty}^r(\mathbb{M}^d)}\left|\int_{\mathbb{M}^d}fd\nu-\sum_{i=1}^n\alpha_kf(x_k)\right|\geq Cn^{-r/d}(\log n)^{-(1+\varepsilon)}, \] where \(C\) is a constant depending only on~\(\varepsilon\).
reviewed by: Szymon Wąsowicz
Mathematics Subject Classification ID: 41A55
Mathematics Subject Classification ID: 41A25
Mathematics Subject Classification ID: 41A63
zbMATH DE Number: 5602668
zbMATH Keywords: cubature
zbMATH Keywords: quadrature
zbMATH Keywords: approximate integration
zbMATH Keywords: rate of convergence
author: Alexander Kushpel
MaRDI profile type: Publication
full work available at URL: j.jfa.2009.04.018
OpenAlex ID: W2083219623
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Wikidata QID: Q125784005
DOI: 10.1016/J.JFA.2009.04.018
description: scientific article; zbMATH DE number 5602668