Harmonic analysis in phase space / by Gerald B. Folland.

Format:

Book

Author/Creator:

Folland, G. B.

Series:

Annals of mathematics studies ; no. 122.

Annals of mathematics studies ; no. 122

Language:

English

Subjects (All):

Phase space (Statistical physics).

Harmonic analysis.

Physical Description:

ix, 277 pages ; 24 cm.

Place of Publication:

Princeton, N.J. : Princeton University Press, 1989.

Summary:

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.

Contents:

Prologue: Some Matters of Notation 3

Chapter 1. The Heisenberg Group and Its Representations 9

1. Background from physics 9

Hamiltonian mechanics 10

Quantum mechanics 12

Quantization 15

2. The Heisenberg group 17

The automorphisms of the Heisenberg group 19

3. The Schrodinger representation 21

The integrated representation 23

Twisted convolution 25

The uncertainty principle 27

4. The Fourier-Wigner transform 30

Radar ambiguity functions 33

5. The Stone-von Neumann theorem 35

The group Fourier transform 37

6. The Fock-Bargmann representation 39

Some motivation and history 47

7. Hermite functions 51

8. The Wigner transform 56

9. The Laguerre connection 63

10. The nilmanifold representation 68

Chapter 2. Quantization and Pseudodifferential Operators 78

1. The Weyl correspondence 79

Covariance properties 83

Symbol classes 86

2. The Kohn-Nirenberg correspondence 93

3. The product formula 103

4. Basic pseudodifferential theory 111

Wave front sets 118

5. The Calderon-Vaillancourt theorems 121

6. The sharp Garding inequality 129

7. The Wick and anti-Wick correspondences 137

Chapter 3. Wave Packets and Wave Fronts 143

1. Wave packet expansions 144

2. A characterization of wave front sets 154

3. Analyticity and the FBI transform 159

4. Gabor expansions 164

Chapter 4. The Metaplectic Representation 170

1. Symplectic linear algebra 170

2. Construction of the metaplectic representation 177

The Fock model 180

3. The infinitesimal representation 185

4. Other aspects of the metaplectic representation 191

Integral formulas 191

Irreducible subspaces 194

Dependence on Planck's constant 195

The extended metaplectic representation 196

The Groenewold-van Hove theorems 197

Some applications 199

5. Gaussians and the symmetric space 200

Characterizations of Gaussians 206

6. The disc model 210

7. Variants and analogues 216

Restrictions of the metaplectic representation 216

U(n,n) as a complex symplectic group 217

The spin representation 220

Chapter 5. The Oscillator Semigroup 223

1. The Schrodinger model 223

The extended oscillator semigroup 234

2. The Hermite semigroup 236

3. Normalization and the Cayley transform 239

4. The Fock model 246

Appendix A. Gaussian Integrals and a Lemma on Determinants 256

Appendix B. Some Hilbert Space Results 260.

Notes:

Includes index.

Bibliography: pages [265]-274.

ISBN:

0691085277 :

0691085285

OCLC:

18681668