Harmonic Analysis in Phase Space. (AM-122)

Harmonic Analysis in Phase Space. (AM-122)

by Gerald B. Folland, G. B. Folland
     
 

This book provides the first coherent account of the area of analysis that involves the Heisenberg group, quantization, the Weyl calculus, the metaplectic representation, wave packets, and related concepts. This circle of ideas comes principally from mathematical physics, partial differential equations, and Fourier analysis, and it illuminates all these subjects.

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Overview

This book provides the first coherent account of the area of analysis that involves the Heisenberg group, quantization, the Weyl calculus, the metaplectic representation, wave packets, and related concepts. This circle of ideas comes principally from mathematical physics, partial differential equations, and Fourier analysis, and it illuminates all these subjects. The principal features of the book are as follows: a thorough treatment of the representations of the Heisenberg group, their associated integral transforms, and the metaplectic representation; an exposition of the Weyl calculus of pseudodifferential operators, with emphasis on ideas coming from harmonic analysis and physics; a discussion of wave packet transforms and their applications; and a new development of Howe's theory of the oscillator semigroup.

Product Details

ISBN-13:
9780691085289
Publisher:
Princeton University Press
Publication date:
03/01/1989
Series:
Annals of Mathematics Studies Series, #12
Edition description:
New Edition
Pages:
288
Product dimensions:
5.90(w) x 8.90(h) x 0.80(d)

Meet the Author

Table of Contents

Prefacevii
Prologue: Some Matters of Notation3
Chapter 1.The Heisenberg Group and Its Representations9
1.Background from physics9
Hamiltonian mechanics10
Quantum mechanics12
Quantization15
2.The Heisenberg group17
The automorphisms of the Heisenberg group19
3.The Schrodinger representation21
The integrated representation23
Twisted convolution25
The uncertainty principle27
4.The Fourier-Wigner transform30
Radar ambiguity functions33
5.The Stone-von Neumann theorem35
The group Fourier transform37
6.The Fock-Bargmann representation39
Some motivation and history47
7.Hermite functions51
8.The Wigner transform56
9.The Laguerre connection63
10.The nilmanifold representation68
11.Postscripts73
Chapter 2.Quantization and Pseudodifferential Operators78
1.The Weyl correspondence79
Covariance properties83
Symbol classes86
Miscellaneous remarks and examples90
2.The Kohn-Nirenberg correspondence93
3.The product formula103
4.Basic pseudodifferential theory111
Wave front sets118
5.The Calderon-Vaillancourt theorems121
6.The sharp Garding inequality129
7.The Wick and anti-Wick correspondences137
Chapter 3.Wave Packets and Wave Fronts143
1.Wave packet expansions144
2.A characterization of wave front sets154
3.Analyticity and the FBI transform159
4.Gabor expansions164
Chapter 4.The Metaplectic Representation170
1.Symplectic linear algebra170
2.Construction of the metaplectic representation177
The Fock model180
3.The infinitesimal representation185
4.Other aspects of the metaplectic representation191
Integral formulas191
Irreducible subspaces194
Dependence on Planck's constant195
The extended metaplectic representation196
The Groenewold-van Hove theorems197
Some applications199
5.Gaussians and the symmetric space200
Characterizations of Gaussians206
6.The disc model210
7.Variants and analogues216
Restrictions of the metaplectic representation216
U(n,n) as a complex symplectic group217
The spin representation220
Chapter 5.The Oscillator Semigroup223
1.The Schrodinger model223
The extended oscillator semigroup234
2.The Hermite semigroup236
3.Normalization and the Cayley transform239
4.The Fock model246
Appendix A.Gaussian Integrals and a Lemma on Determinants256
Appendix B.Some Hilbert Space Results260
Bibliography265
Index275

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