Polynomial interpolation and hyperinterpolation over general regions
type: Item
instance of: scholarly article
title: Polynomial interpolation and hyperinterpolation over general regions
zbMATH Open document ID: 0839.41006
author: Ian H. Sloan
published in: Journal of Approximation Theory
publication date: 1995-12-20T00:00:00Z
review text: This paper studies a generalization of polynomial interpolation: given a continuous function over a rather general manifold, hyperinterpolation is a linear approximation that makes use of values of \(f\) on a well chosen finite set. The approximation is a discrete least-squares approximation constructed with the aid of a high-order quadrature rule: the role of the quadrature rule is to approximate the Fourier coefficients of \(f\) with respect to an orthonormal basis of the space of polynomials of degree \(\leq n\). The principal result is a generalization of the result of Erdös and Turán for classical interpolation at the zeros of orthogonal polynomials: for a rule of suitably high order (namely \(2n\) or greater), the \(L_2\) error of the approximation is shown to be within a constant factor of the error of best uniform approximation by polynomials of degree \(\leq n\). The \(L_2\) error therefore converges to zero as the degree of the approximating polynomial approach \(\infty\). As example discussed in detail is the approximation of continuous functions on the sphere in \(\mathbb{R}^s\) by spherical polynomials.
Mathematics Subject Classification ID: 41A10
Mathematics Subject Classification ID: 41A55
Mathematics Subject Classification ID: 65D32
zbMATH DE Number: 827935
zbMATH Keywords: multivariate polynomial approximation
zbMATH Keywords: quadrature
MaRDI profile type: Publication
full work available at URL: jath.1995.1119
OpenAlex ID: W1998258828
DOI: 10.1006/JATH.1995.1119
description: scientific article; zbMATH DE number 827935