Harmonic Analysis in Phase Space

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.

Preface	p. vii
Prologue: Some Matters of Notation	p. 3
The Heisenberg Group and Its Representations	p. 9
Background from physics	p. 9
Hamiltonian mechanics	p. 10
Quantum mechanics	p. 12
Quantization	p. 15
The Heisenberg group	p. 17
The automorphisms of the Heisenberg group	p. 19
The Schrodinger representation	p. 21
The integrated representation	p. 23
Twisted convolution	p. 25
The uncertainty principle	p. 27
The Fourier-Wigner transform	p. 30
Radar ambiguity functions	p. 33
The Stone-von Neumann theorem	p. 35
The group Fourier transform	p. 37
The Fock-Bargmann representation	p. 39
Some motivation and history	p. 47
Hermite functions	p. 51
The Wigner transform	p. 56
The Laguerre connection	p. 63
The nilmanifold representation	p. 68
Postscripts	p. 73
Quantization and Pseudodifferential Operators	p. 78
The Weyl correspondence	p. 79
Covariance properties	p. 83
Symbol classes	p. 86
Miscellaneous remarks and examples	p. 90
The Kohn-Nirenberg correspondence	p. 93
The product formula	p. 103
Basic pseudodifferential theory	p. 111
Wave front sets	p. 118
The Calderon-Vaillancourt theorems	p. 121
The sharp Garding inequality	p. 129
The Wick and anti-Wick correspondences	p. 137
Wave Packets and Wave Fronts	p. 143
Wave packet expansions	p. 144
A characterization of wave front sets	p. 154
Analyticity and the FBI transform	p. 159
Gabor expansions	p. 164
The Metaplectic Representation	p. 170
Symplectic linear algebra	p. 170
Construction of the metaplectic representation	p. 177
The Fock model	p. 180
The infinitesimal representation	p. 185
Other aspects of the metaplectic representation	p. 191
Integral formulas	p. 191
Irreducible subspaces	p. 194
Dependence on Planck's constant	p. 195
The extended metaplectic representation	p. 196
The Groenewold-van Hove theorems	p. 197
Some applications	p. 199
Gaussians and the symmetric space	p. 200
Characterizations of Gaussians	p. 206
The disc model	p. 210
Variants and analogues	p. 216
Restrictions of the metaplectic representation	p. 216
U(n,n) as a complex symplectic group	p. 217
The spin representation	p. 220
The Oscillator Semigroup	p. 223
The Schrodinger model	p. 223
The extended oscillator semigroup	p. 234
The Hermite semigroup	p. 236
Normalization and the Cayley transform	p. 239
The Fock model	p. 246
Gaussian Integrals and a Lemma on Determinants	p. 256
Some Hilbert Space Results	p. 260
Bibliography	p. 265
Index	p. 275
Table of Contents provided by Syndetics. All Rights Reserved.

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Harmonic Analysis in Phase Space

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