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Harmonic Analysis in Phase Space. (AM-122), Volume 122 : Annals of Mathematics Studies (Paperback) - Gerald B. Folland

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

Annals of Mathematics Studies (Paperback)

Paperback

Published: 21st March 1989
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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.

"[This book] is a valiant attempt to present an account of [harmonic analysis in phase space], with an emphasis on the analysis-quantum mechanics and pseudodifferential operators... The author has taken great pains to express himself clearly and ... the notation is consistent throughout... The author should be congratulated on a very valuable addition to the library of harmonic analysis."--Zentralblatt fur Mathematik

Prefacep. vii
Prologue: Some Matters of Notationp. 3
The Heisenberg Group and Its Representationsp. 9
Background from physicsp. 9
Hamiltonian mechanicsp. 10
Quantum mechanicsp. 12
Quantizationp. 15
The Heisenberg groupp. 17
The automorphisms of the Heisenberg groupp. 19
The Schrodinger representationp. 21
The integrated representationp. 23
Twisted convolutionp. 25
The uncertainty principlep. 27
The Fourier-Wigner transformp. 30
Radar ambiguity functionsp. 33
The Stone-von Neumann theoremp. 35
The group Fourier transformp. 37
The Fock-Bargmann representationp. 39
Some motivation and historyp. 47
Hermite functionsp. 51
The Wigner transformp. 56
The Laguerre connectionp. 63
The nilmanifold representationp. 68
Postscriptsp. 73
Quantization and Pseudodifferential Operatorsp. 78
The Weyl correspondencep. 79
Covariance propertiesp. 83
Symbol classesp. 86
Miscellaneous remarks and examplesp. 90
The Kohn-Nirenberg correspondencep. 93
The product formulap. 103
Basic pseudodifferential theoryp. 111
Wave front setsp. 118
The Calderon-Vaillancourt theoremsp. 121
The sharp Garding inequalityp. 129
The Wick and anti-Wick correspondencesp. 137
Wave Packets and Wave Frontsp. 143
Wave packet expansionsp. 144
A characterization of wave front setsp. 154
Analyticity and the FBI transformp. 159
Gabor expansionsp. 164
The Metaplectic Representationp. 170
Symplectic linear algebrap. 170
Construction of the metaplectic representationp. 177
The Fock modelp. 180
The infinitesimal representationp. 185
Other aspects of the metaplectic representationp. 191
Integral formulasp. 191
Irreducible subspacesp. 194
Dependence on Planck's constantp. 195
The extended metaplectic representationp. 196
The Groenewold-van Hove theoremsp. 197
Some applicationsp. 199
Gaussians and the symmetric spacep. 200
Characterizations of Gaussiansp. 206
The disc modelp. 210
Variants and analoguesp. 216
Restrictions of the metaplectic representationp. 216
U(n,n) as a complex symplectic groupp. 217
The spin representationp. 220
The Oscillator Semigroupp. 223
The Schrodinger modelp. 223
The extended oscillator semigroupp. 234
The Hermite semigroupp. 236
Normalization and the Cayley transformp. 239
The Fock modelp. 246
Gaussian Integrals and a Lemma on Determinantsp. 256
Some Hilbert Space Resultsp. 260
Bibliographyp. 265
Indexp. 275
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780691085289
ISBN-10: 0691085285
Series: Annals of Mathematics Studies (Paperback)
Audience: Tertiary; University or College
Format: Paperback
Language: English
Number Of Pages: 288
Published: 21st March 1989
Country of Publication: US
Dimensions (cm): 23.44 x 15.6  x 1.83
Weight (kg): 0.41