Harmonic Analysis in Phase Space
by Folland, Gerald B.Format: Paperback
Pub. Date: 1989-03-01
Publisher(s): Princeton Univ Pr
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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.
Table of Contents
| 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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