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9 - Through the Kaleidoscope: Symmetries, Groups and Chebyshev-Approximations from a Computational Point of View

Published online by Cambridge University Press:  05 December 2012

By
H. Munthe-Kaas ,
M. Nome  and
B. N. Ryland
Edited by
Felipe Cucker ,
Teresa Krick ,
Allan Pinkus  and
Agnes Szanto
H. Munthe-Kaas
Affiliation:
University of Bergen
M. Nome
Affiliation:
University of Bergen
B. N. Ryland
Affiliation:
University of Bergen
Felipe Cucker
Affiliation:
City University of Hong Kong
Teresa Krick
Affiliation:
Universidad de Buenos Aires, Argentina
Allan Pinkus
Affiliation:
Technion - Israel Institute of Technology, Haifa
Agnes Szanto
Affiliation:
North Carolina State University
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Summary

Abstract

In this paper we survey parts of group theory, with emphasis on structures that are important in design and analysis of numerical algorithms and in software design. In particular, we provide an extensive introduction to Fourier analysis on locally compact abelian groups, and point towards applications of this theory in computational mathematics. Fourier analysis on non-commutative groups, with applications, is discussed more briefly. In the final part of the paper we provide an introduction to multivariate Chebyshev polynomials. These are constructed by a kaleidoscope of mirrors acting upon an abelian group, and have recently been applied in numerical Clenshaw-Curtis type numerical quadrature and in spectral element solution of partial differential equations, based on triangular and simplicial subdivisions of the domain.

Introduction

Group theory is the mathematical language of symmetry. As a mature branch of mathematics, with roots going almost two centuries back, it has evolved into a highly technical discipline. Many texts on group theory and representation theory are not readily accessible to applied mathematicians and computational scientists, and the relevance of group theoretical techniques in computational mathematics is not widely recognized.

Nevertheless, it is our conviction that knowledge of central parts of group theory and harmonic analysis on groups is invaluable also for computational scientists, both as a language to unify, analyze and generalize computational algorithms and also as an organizing principle of mathematical software construction.

Type
Chapter
Information
Foundations of Computational Mathematics, Budapest 2011 , pp. 188 - 229
Publisher: Cambridge University Press
Print publication year: 2012

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References

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