Spherical Harmonics and Approximations on the Unit Sphere: An 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
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2044
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Kendall Atkinson
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Weimin Han
Spherical Harmonics and Approximations on the Unit Sphere: An Introduction
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Kendall Atkinson University of Iowa Department of Mathematics and Department of Computer Science Iowa City, IA 52242 USA
Weimin Han University of Iowa Department of Mathematics Iowa City, IA 52242 USA
ISBN 978-3-642-25982-1 e-ISBN 978-3-642-25983-8 DOI 10.1007/978-3-642-25983-8 Springer Heidelberg Dordrecht London New York Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2012931330 Mathematics Subject Classification (2010): 41A30, 65N30, 65R20 c Springer-Verlag Berlin Heidelberg 2012 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Preface
Spherical harmonics have been studied extensively and applied to solving a wide range of problems in the sciences and engineering. Interest in approximations and numerical methods for problems over spheres has grown steadily. These notes provide an introduction to the theory of spherical harmonics in an arbitrary dimension as well as a summarizing account of classical and recent results on some aspects of approximation by spherical polynomials and numerical integration over the 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 sphere, especially on the unit sphere S2 in R3 . We also discuss briefly some related work for approximation on the unit disk in R2 , with those results being generalizable to the unit ball in more dimensions. The subject of theoretical approximation of functions on Sd , d > 2, using spherical polynomials has been an active area of research over the past several decades. We summarize some of the major results, giving some insight into them; however, these notes are not intended to be a complete development of the theory of approximation of functions on Sd by spherical polynomials. There are a number of other approaches to the approximation of functions on the spher
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