Spherical harmonics and approximations on the unit sphere. An introduction
type: Item
instance of: scholarly article
title: Spherical harmonics and approximations on the unit sphere. An introduction
zbMATH Open document ID: 1254.41015
DOI: 10.1007/978-3-642-25983-8
author: Weimin Han
author: Kendall E. Atkinson
published in: Lecture Notes in Mathematics
publication date: 2012-01-02T00:00:00Z
review text: The book concentrates on the theory of spherical harmonics on the unit sphere of a general \(d\)-dimensional Euclidian space. It summarizes the results related to Legendre and Gegenbauer polynomials as well as the theory of differentiation and integration over the \(d\)-dimensional unit sphere and the associated function spaces. Applications of such theoretical constructions to the theory of approximation and numerical quadrature together with description of spectral methods for several well known problems are presented in the remaining part of the book. The style of material presentation was deliberately chosen by the authors with the aim to make the theory described in the book accessible to a wider audience of readers with only some basic knowledge in the functional analysis and measure theory. The book consists of a preface, six chapters, a references list and an index. In Chapter 1, the authors supply the reader with some basic notation and facts. Chapter 2 presents the theory of spherical harmonics from the point of view of the linear function space defined on a unit sphere \(S^d\). Spherical harmonics are introduced as a restriction of harmonic homogeneous polynomials to the sphere. After some basic definitions stated at the beginning of this chapter the authors move on to a description of several important properties of the spherical harmonics such as an addition theorem and the Funk-Hecke formula. And then they define the projection operator onto spherical harmonics function subspaces, which proves to be a useful tool in proving various properties of the spherical harmonics and establishing relations between several polynomial spaces. These topics are discussed later in the chapter. After that the authors move to the properties and representation formulas for Legendre and Gegenbauer polynomials. The rest of the chapter is devoted to the discussion of associated Legendre functions and their role in generating orthogonal bases for spherical harmonic function …
Mathematics Subject Classification ID: 41A30
Mathematics Subject Classification ID: 33C55
Mathematics Subject Classification ID: 65D30
Mathematics Subject Classification ID: 65R20
zbMATH DE Number: 5994018
zbMATH Keywords: spherical harmonics
zbMATH Keywords: Legendre polinomials
zbMATH Keywords: Gegenbauer polynomials
zbMATH Keywords: Funk-Hecke formula
zbMATH Keywords: approximation theory
zbMATH Keywords: quadrature
zbMATH Keywords: Galerkin method
zbMATH Keywords: unit sphere
reviewed by: Vladimir L. Makarov
describes a project that uses: STRIPACK
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-642-25983-8
OpenAlex ID: W167165788
description: scientific article; zbMATH DE number 5994018