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© 2012

Spherical Harmonics and Approximations on the Unit Sphere: An Introduction

  • An easily accessible introduction to the theory of spherical harmonics in an arbitrary dimension

  • A summarizing account of classical and recent results on some aspects of function approximations by spherical polynomials and numerical integration over the unit sphere

  • Useful for graduate students and researchers interested in solving problems over the sphere

  • Good for a graduate level topic course on spherical harmonics and approximations over the sphere

Book

Part of the Lecture Notes in Mathematics book series (LNM, volume 2044)

Also part of the Ecole d'Eté Probabilit.Saint-Flour book sub series (volume 2044)

Table of contents

  1. Front Matter
    Pages i-ix
  2. Kendall Atkinson, Weimin Han
    Pages 1-9
  3. Kendall Atkinson, Weimin Han
    Pages 11-86
  4. Kendall Atkinson, Weimin Han
    Pages 87-130
  5. Kendall Atkinson, Weimin Han
    Pages 131-163
  6. Kendall Atkinson, Weimin Han
    Pages 165-210
  7. Kendall Atkinson, Weimin Han
    Pages 211-236
  8. Back Matter
    Pages 237-244

About this book

Introduction

These notes provide an introduction to the theory of spherical harmonics in an arbitrary dimension as well as
an overview of classical and recent results on some aspects of the approximation of functions by spherical polynomials and numerical integration over the unit sphere. The notes are intended for graduate students in the mathematical sciences and researchers who are interested in solving problems involving partial differential and integral equations on the unit sphere, especially on the unit sphere in three-dimensional Euclidean space. Some related work for approximation on the unit disk in the plane is also briefly discussed, with results being generalizable to the unit ball in more dimensions.

Keywords

41A30, 65N30, 65R20 approximation theory quadrature spherical harmonic

Authors and affiliations

  1. 1.Department of Mathematics &, Department of Computer ScienceUniversity of IowaIowa CityUSA
  2. 2.Department of MathematicsUniversity of Iowa CityIowa CityUSA

Bibliographic information

Industry Sectors
Energy, Utilities & Environment
Engineering

Reviews

From the reviews:

“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. … The style of material presentation … 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.” (Vladimir L. Makarov, Zentralblatt MATH, Vol. 1254, 2013)

“This is a very well-written, self-contained monograph on spherical harmonics. It is an excellent reference source for researchers and graduate students who are interested in polynomial approximation, numerical integration, differentiation and solution of partial differential and integral equations over the sphere.” (Feng Dai, Mathematical Reviews, January, 2013)