Harmonic Analysis in Phase Space. (AM-122), Volume 122

✍ Scribed by Gerald B. Folland

• Publisher: Princeton University Press
• Year: 2016
• Tongue: English
• Leaves: 288
• Series: Annals of Mathematics Studies; 122
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

CONTENTS
PREFACE
Prologue. Some Matters of Notation
CHAPTER 1. THE HEISENBERG GROUP AND ITS REPRESENTATIONS
1. Background from physics
Hamiltonian mechanics
Quantum mechanics
Quantization
2. The Heisenberg group
The automorphisms of the Heisenberg group
3. The Schrödinger representation
The integrated representation
Twisted convolution
The uncertainty principle
4. The Fourier–Wigner transform
Radar ambiguity functions
5. The St one–von Neumann theorem
The group Fourier transform
6. The Fock–Bargmann representation
Some motivation and history
7. Hermite functions
8. The Wigner transform
9. The Laguerre connection
10. The nilmanifold representation
11. Postscripts
CHAPTER 2. QUANTIZATION AND PSEUDODIFFERENTIAL OPERATORS
1. The Weyl correspondence
Covariance properties
Symbol classes
Miscellaneous remarks and examples
2. The Kohn–Nirenberg correspondence
3. The product formula
4. Basic pseudodifferential theory
Wave front sets
5. The Calderón–Vaillancourt theorems
6. The sharp Gårding inequality
7. The Wick and anti-Wick correspondences
CHAPTER 3. WAVE PACKETS AND WAVE FRONTS
1. Wave packet expansions
2. A characterization of wave front sets
3. Analyticity and the FBI transform
4. Gabor expansions
CHAPTER 4. THE METAPLECTIC REPRESENTATION
1. Symplectic linear algebra
2. Construction of the metaplectic representation
The Fock model
3. The infinitesimal representation
4. Other aspects of the metaplectic representation
Integral formulas
Irreducible subspaces
Dependence on Planck’s constant
The extended metaplectic representation
The Groenewold–van Hove theorems
Some applications
5. Gaussians and the symmetric space
Characterizations of Gaussians
6. The disc model
7. Variants and analogues
Restrictions of the metaplectic representation
U(n,n) as a complex symplectic group
The spin representation
CHAPTER 5. THE OSCILLATOR SEMIGROUP
1. The Schrödinger model
The extended oscillator semigroup
2. The Hermite semigroup
3. Normalization and the Cayley transform
4. The Fock model
Appendix A. Gaussian Integrals and a Lemma on Determinants
Appendix B. Some Hilbert Space Results
Bibliography
Index