# Hyperinterpolation on the sphere

Hesse, Kerstin and Sloan, Ian H (2006) Hyperinterpolation on the sphere. In: Govil, N K, Mhaskar, H N, Mohapatra, Ram N, Nashed, Zuhair and Szabados, J (eds.) Frontiers in Interpolation and Approximation (Dedicated to the memory of Ambikeshwar Sharma). Chapman & Hall/CRC, pp. 213-248. ISBN 978-1-584-88636-5

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## Abstract

In this paper we survey hyperinterpolation on the sphere $\\mathbb{S}^d$, $d\\geq 2$. The hyperinterpolation operator $L_n$ is a linear projection onto the space $\\mathbb{P}_n(\\mathbb{S}^d)$ of spherical polynomials of degree $\\leq n$, which is obtained from $L_2(\\mathbb{S}^d)$-orthogonal projection onto $\\mathbb{P}_n(\\mathbb{S}^d)$ by discretizing the integrals in the $L_2(\\mathbb{S}^d)$ inner products by a positive-weight numerical integration rule of polynomial degree of exactness $2n$. Thus hyperinterpolation is a kind of `discretized orthogonal projection' onto $\\mathbb{P}_n(\\mathbb{S}^d)$, which is relatively easy and inexpensive to compute. In contrast, the $L_2(\\mathbb{S}^d)$-orthogonal projection onto $\\mathbb{P}_n(\\mathbb{S}^d)$ cannot generally be computed without some discretization of the integrals in the inner products; hyperinterpolation is a realization of such a discretization. We compare hyperinterpolation with $L_2(\\mathbb{S}^d)$-orthogonal projection onto $\\mathbb{P}_n(\\mathbb{S}^d)$ and with polynomial interpolation onto $\\mathbb{P}_n(\\mathbb{S}^d)$: we discuss the properties, estimates of the operator norms in terms of $n$, and estimates of the approximation error. We also present a new estimate of the approximation error of hyperinterpolation in the Sobolev space setting, that is, $L_n:H^t(\\mathbb{S}^d)\\rightarrow H^s(\\mathbb{S}^d)$, with $t\\geq s\\geq 0$ and $t&gt;d/2$, where $H^s(\\mathbb{S}^d)$ is for integer $s$ roughly the Sobolev space of those functions whose generalized derivatives up to the order $s$ are square-integrable.

Item Type: Book Section School of Mathematical and Physical Sciences > Mathematics Kerstin Hesse 06 Feb 2012 18:48 11 Apr 2012 11:43 http://sro.sussex.ac.uk/id/eprint/18442