A theorem concerning approximation on the sphere
type: Item
instance of: scholarly article
title: A theorem concerning approximation on the sphere
zbMATH Open document ID: 0565.41033
DOI: 10.1007/BF01989806
author: P. I. Lizorkin
author: S. M. Nikol'skii
published in: Analysis Mathematica
publication date: 1983-01-01T00:00:00Z
review text: The authors prove the following interesting theorem concerning approximation on a unit sphere \(\sigma\) in \(R^ n\). Theorem: If \(f(\mu)\in H^ r_ p(\sigma)\), then \(\| f-T_{\nu}\|_{L_ p(\sigma)}\leq (C\| f\| /\nu^ r)H^ r_ p(\sigma)\) \((\nu =1,2,...)\) for a suitable sequence \(\{T_{\nu}(\mu)\}\) of spherical polynomials of order \(\nu\), where c is independent of f and \(\nu\). Conversely, if f(\(\mu)\), \(\mu\in \sigma\), is a function such that for some sequence \(\{T_{\nu}(\mu)\}\) of spherical polynomials of order \(\nu\), \(\| f-T_{\nu}\|_{L_ p(\sigma)}\leq K/\nu^ r\) \((\nu =1,2,...)\) then \(f\in H^ r_ p(\sigma)\) and \(\| f\|_{H^ r_ p(\sigma)}\leq C_ 1(\| f\|_{L_ p(\sigma)}+K),\) where K is a constant independent of \(\nu\) and \(C_ 1\) is independent of f and K.
reviewed by: S. M. Mazhar
Mathematics Subject Classification ID: 41A65
Mathematics Subject Classification ID: 41A25
Mathematics Subject Classification ID: 41A63
zbMATH DE Number: 3901386
zbMATH Keywords: Laplace-Beltrami operator
zbMATH Keywords: approximation on a unit sphere \(\sigma \)
zbMATH Keywords: spherical polynomials
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01989806
OpenAlex ID: W1749663818
description: scientific article; zbMATH DE number 3901386