Discretized Newman-Shapiro operators and Jackson's inequality on the sphere
type: Item
instance of: scholarly article
title: Discretized Newman-Shapiro operators and Jackson's inequality on the sphere
zbMATH Open document ID: 0957.41007
DOI: 10.1007/BF03322120
author: Manfred Reimer
published in: Results in Mathematics
publication date: 2001-03-22T00:00:00Z
review text: The author takes a quadrature formula on the \(r\)-dimensional sphere, \(r>2\), that has positive weights and which is exact for spherical polynomials of degree \(2\nu+1\), and applies it to what the author refers to as the Newman-Shapiro operator on that sphere. Denoting the resulting operator \(L_\mu\), where \(\mu=2\nu\), he then proves that there exists an absolute constant \(k\) depending only on the dimension \(r\), such that for every function \(F\in C(S^{r-1})\), we have \(\|F-L_\mu F\|\leq k\omega_F(1/\mu)\), where \(\|\cdot\|\) denotes the uniform norm on the sphere and \(\omega_F(t):=\sup\{|F(x)-F(y)|:x,y\in S^{r-1},x\cdot y\geq\cos t\}\). It should be stressed that \(k\) is independent of the quadrature formula provided it possesses the above requirements. The proof involves properties of the Gegenbauer polynomials.
Mathematics Subject Classification ID: 41A17
Mathematics Subject Classification ID: 41A63
Mathematics Subject Classification ID: 65G99
Mathematics Subject Classification ID: 41A05
zbMATH DE Number: 1383728
zbMATH Keywords: approximation on the sphere
zbMATH Keywords: Jackson estimates
zbMATH Keywords: hyperinterpolation
reviewed by: D. Leviatan
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1383728