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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

  • Includes supplementary material: sn.pub/extras

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

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Table of contents (6 chapters)

  1. Front Matter

    Pages i-ix
  2. Preliminaries

    • Kendall Atkinson, Weimin Han
    Pages 1-9
  3. Spherical Harmonics

    • Kendall Atkinson, Weimin Han
    Pages 11-86
  4. Differentiation and Integration over the Sphere

    • Kendall Atkinson, Weimin Han
    Pages 87-130
  5. Approximation Theory

    • Kendall Atkinson, Weimin Han
    Pages 131-163
  6. Numerical Quadrature

    • Kendall Atkinson, Weimin Han
    Pages 165-210
  7. Applications: Spectral Methods

    • Kendall Atkinson, Weimin Han
    Pages 211-236
  8. Back Matter

    Pages 237-244

About this book

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
  • partial differential equations

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)

Authors and Affiliations

  • Department of Mathematics &, Department of Computer Science, University of Iowa, Iowa City, USA

    Kendall Atkinson

  • Department of Mathematics, University of Iowa City, Iowa City, USA

    Weimin Han

Bibliographic Information

Buying options

eBook USD 44.99
Price excludes VAT (Canada)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book USD 59.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions