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The Scope and History of Commutative and Noncommutative Harmonic Analysis
G. W. Mackey
A co-publication of the AMS and the London Mathematical Society.
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History of Mathematics
1992; 370 pp; softcover
Volume: 5
ISBN-10: 0-8218-3790-7
ISBN-13: 978-0-8218-3790-0
List Price: US$64
Member Price: US$51.20
Order Code: HMATH/5.S
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"When I was invited to speak at the conference on the history of analysis given at Rice University [in 1977], I decided that it might be interesting to review the history of mathematics and physics in the last three hundred years or so with heavy emphasis on those parts in which harmonic analysis had played a decisive or at least a major role. I was pleased and somewhat astonished to find how much of both subjects could be included under this rubric ... The picture that gradually emerged as the various details fell into place was one that I found very beautiful, and the process of seeing it do so left me in an almost constant state of euphoria. I would like to believe that others can be led to see this picture by reading my paper, and to facilitate this I have included a large number of short expositions of topics which are not widely understood by non-specialists."

--from the Preface

This volume, containing the paper mentioned above as well as five other reprinted papers by Mackey, presents a sweeping view of the importance, utility, and beauty of harmonic analysis and its connections to other areas of mathematics and science. A seventh paper, written exclusively for this volume, attempts to unify certain themes that emerged after major discoveries in 1967 and 1968 in the areas of Lie algebras, strong interaction physics, statistical mechanics, and nonlinear partial differential equations--discoveries that may at first glance appear to be independent, but which are in fact deeply interrelated.

Reviews

"Expanded versions of lectures given at various conferences ... nice to have them together in one book ... they constitute historic and expository material of the highest quality and importance."

-- Mathematical Reviews

"A remarkable account, due to a first-rank contributor in harmonic analysis, on the work of H. Weyl, one of the outstanding mathematicians of the beginning of the XXth century ... each mathematician interested in the development of mathematical ideas should complement his information by reading the book under review."

-- Zentralblatt MATH

Table of Contents

  • Introduction
  • Harmonic analysis as the exploitation of symmetry: A historical survey
  • Herman Weyl and the application of group theory to quantum mechanics
  • The significance of invariant measures for harmonic analysis
  • Weyl's program and modern physics
  • Induced representations and the applications of harmonic analysis
  • Von Neumann and the early days of ergodic theory
  • Final remarks
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