Generalized hyperinterpolation on the sphere and the Newman-shapiro operators
type: Item
instance of: scholarly article
title: Generalized hyperinterpolation on the sphere and the Newman-shapiro operators
zbMATH Open document ID: 1002.41016
author: Manfred Reimer
published in: Constructive Approximation
publication date: 2002-06-27T00:00:00Z
review text: In this interesting paper, the author modifies Ian Sloan's idea of hyperinterpolation on the sphere, to generalized hyperinterpolation. The advantage is that while hyperinterpolation diverges in the uniform norm, generalized hyperinterpolation converges in the uniform norm. Let \(r\geq 3\) and let \(S^{r-1}\) denote the unit sphere in \(\mathbb{R}^3\). Let \({\mathbf P}^r_\mu\) denote the polynomials of degree \(\leq\mu\) in \(r\) variables, restricted to the unit sphere. Let \(G_\mu\) denote the reproducing kernel for the homogeneous harmonic polynomials of degree \(\mu\) on \(S^{r-1}\). It admits a representation in terms of ultraspherical polynomials. Both hyperinterpolation and generalized hyperinterpolation are based on a sequence of quadrature rules on the sphere, \[ Q_\mu F=\sum^M_{j=1} A_jF(t_j), \] where \(F\) is continuous on \(S^{r-1}\), the nodes \(\{t_j\}\) belong to \(S^{r-1}\), and quite often the weights \(A_j\) are positive. If \[ Q_\mu F=\int_{S^{r-1}} Fd\omega\text{ for all }F\in {\mathbf P}^r_\mu, \] where \(\omega\) is a Lebesgue measure on \(S^{r-1}\), the rules are said to have exact degree \(\mu+1\). Generalized hyperinterpolation also involves an infinite matrix \(A=(a_{j,k})\) satisfying the following conditions: (i) \(a_{j,k} =0\) for \(k>j\); (ii) \(\lim_{j\to\infty} a_{j,k}=1\) for \(k\in\{0,1\}\); (iii) \(K_\mu= \sum^\mu_{k=0} a_{\mu,k} G_k\geq 0\) in \([-1,1]\). One may think of \(K_\mu\) as a smoothing of the reproducing kernel \(G_\mu\), for example a Cesàro mean. Now one can define the generalized hyperinterpolation operator \(L_\mu\) based on \(Q_\mu\) by \[ L_\mu F(t)= \sum^M_{j=0} A_jF(t_j) K_\mu(t_j\cdot t). \] The author proves that \(\{L_\mu\}\) are positive operators that converge in the uniform norm as \(\mu\to \infty\) for all \(F\) continuous on \(S^{r-1}\). As examples of the matrix \(A\), the author discusses Cesàro methods, and also the Newman-Shapiro summation method. This paper is of great interest to anyone working i…
Mathematics Subject Classification ID: 41A63
Mathematics Subject Classification ID: 41A10
zbMATH DE Number: 1761095
zbMATH Keywords: Newman-Shapiro operators
zbMATH Keywords: hyperinterpolation
zbMATH Keywords: quadrature rules
reviewed by: Doron S. Lubinsky
MaRDI profile type: Publication
full work available at URL: s00365-001-0008-6
OpenAlex ID: W2038906925
DOI: 10.1007/S00365-001-0008-6
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description: scientific article; zbMATH DE number 1761095