Coherent states, wavelets and their generalizations:
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Springer
2000
|
| Series: | Graduate texts in contemporary physics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIII, 418 S. Ill., graph. Darst. |
| ISBN: | 0387989080 |
Staff View
MARC
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Record in the Search Index
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| adam_text | Contents
Preface v
1 Introduction 1
2 Canonical Coherent States 13
2.1 Minimal uncertainty states 13
2.2 The group theoretical backdrop 17
2.3 Some functional analytic properties 20
2.4 A complex analytic viewpoint (* 24
2.5 Some geometrical considerations 26
2.6 Outlook 27
2.7 Two illustrative examples 27
2.7.1 A quantization problem * 27
2.7.2 An application to atomic physics (*J 29
3 Positive Operator Valued Measures and Frames 33
3.1 Definitions and main properties 34
3.2 The case of a tight frame 40
3.3 Example: A commutative POV measure 41
3.4 Discrete frames 42
4 Some Group Theory 47
4.1 Homogeneous spaces, quasi invariant, and
invariant measures 47
x Contents
4.2 Induced representations and systems of covariance .... 54
4.2.1 Vector coherent states 59
4.2.2 Discrete series representations of SU(1,1) .... 61
4.2.3 The regular representations of a group 67
4.3 An extended Schur s lemma 68
4.4 Harmonic analysis on locally compact abelian groups . . 70
4.4.1 Basic notions 70
4.4.2 Lattices in LCA groups 72
4.4.3 Sampling in LCA groups 73
4.5 Lie groups and Lie algebras: A reminder 75
4.5.1 Lie algebras 75
4.5.2 Lie groups 77
4.5.3 Extensions of Lie algebras and Lie groups .... 82
4.5.4 Contraction of Lie algebras and Lie groups .... 85
5 Hilbert Spaces with Reproducing Kernels
and Coherent States 89
5.1 A motivating example 89
5.2 Measurable fields and direct integrals 92
5.3 Reproducing kernel Hilbert spaces 94
5.3.1 Positive definite kernels and evaluation maps ... 94
5.3.2 Coherent states and POV functions 99
5.3.3 Some isomorphisms, bases, and f selections . . . 101
5.3.4 A reconstruction problem: Example of a
holomorphic map (*) 104
5.4 Some properties of reproducing kernel Hilbert spaces . . 106
6 Square Integrable and Holomorphic Kernels 109
6.1 Square integrable kernels 109
6.2 Holomorphic kernels 112
6.3 Coherent states: The holomorphic case 116
6.3.1 An example of a holomorphic frame (*) 119
6.3.2 A nonholomorphic excursion (*) 122
7 Covariant Coherent States 127
7.1 Covariant coherent states 128
7.1.1 A general definition 128
7.1.2 The Gilmore Perelomov CS and vector CS . . . . 129
7.1.3 A geometrical setting 133
7.2 Example: The classical theory of coherent states .... 136
7.2.1 CS of compact semisimple Lie groups 136
7.2.2 CS of noncompact semisimple Lie groups 139
7.2.3 CS of non semisimple Lie groups 141
7.3 Square integrable covariant CS: The general case .... 141
7.3.1 Some further generalizations 144
Contents xi
8 Coherent States from Square Integrable Representations 147
8.1 Square integrable group representations 148
8.2 Orthogonality relations 155
8.3 The Wigner map 160
8.4 Modular structures and statistical mechanics 164
9 Some Examples and Generalizations 171
9.1 A class of semidirect product groups 171
9.1.1 Three concrete examples ^ 176
9.1.2 A broader setting 179
9.2 A generalization: a and V admissibility 181
9.2.1 Example of the Galilei group * 186
9.2.2 CS of the isochronous Galilei group (* 190
9.2.3 Atomic coherent states 196
10 CS of General Semidirect Product Groups 199
10.1 Squeezed states * 200
10.2 Geometry of semidirect product groups 204
10.2.1 A special class of orbits 204
10.2.2 The coadjoint orbit structure of T 206
10.2.3 Measures on T 209
10.2.4 Induced representations of semidirect products . . 213
10.3 CS of semidirect products 214
10.3.1 Admissible affine sections 220
11 CS of the Relativity Groups 225
11.1 The Poincare groups 7 j(l, 3) and PJ(1,1) 225
11.1.1 The Poincare group in 1+3 dimensions,
7+(l,3) * 225
11.1.2 The Poincare group in 1+1 dimensions,
7 X(1.1) w 235
11.1.3 Poincare CS: The massless case 240
11.2 The Galilei groups £(1,1) and Q = (7(1,3) (* 242
11.3 The anti de Sitter group SOO(1,2) and
its contraction(s) *) 245
12 Wavelets 257
12.1 A word of motivation 257
12.2 Derivation and properties of the 1 D continuous
wavelet transform (*) 261
12.3 A mathematical aside: Extension to distributions .... 266
12.4 Interpretation of the continuous wavelet transform .... 272
12.4.1 The CWT as phase space representation 272
12.4.2 Localization properties and physical interpretation
of the CWT 273
xii Contents
12.5 Discretization of the continuous WT: Discrete frames . . 274
12.6 Ridges and skeletons 276
12.7 Applications 278
13 Discrete Wavelet Transforms 283
13.1 The discrete time or dyadic WT 283
13.1.1 Multiresolution analysis and orthonormal
wavelet bases 284
13.1.2 Connection with filters and the
subband coding scheme 285
13.1.3 Generalizations 287
13.1.4 Applications 288
13.2 Towards a fast CWT: Continuous wavelet packets .... 289
13.3 Wavelets on the finite field Zp *} 290
13.4 Algebraic wavelets 292
13.4.1 r wavelets of Haar on the line *} 292
13.4.2 Pisot wavelets, etc. (* 303
14 Multidimensional Wavelets 307
14.1 Going to higher dimensions 307
14.2 Mathematical analysis (*) 308
14.3 The 2 D case 316
14.3.1 Minimality properties 316
14.3.2 Interpretation, visualization problems,
and calibration 318
14.3.3 Practical applications of the CWT in
two dimensions 322
14.3.4 The discrete WT in two dimensions 327
14.3.5 Continuous wavelet packets in two dimensions . . 330
15 Wavelets Related to Other Groups 331
15.1 Wavelets on the sphere and similar manifolds 331
15.1.1 The two sphere (* 331
15.1.2 Generalization to other manifolds 336
15.2 The affine Weyl Heisenberg group * 338
15.3 The affine or similitude groups of space time 342
15.3.1 Kinematical wavelets (*) 342
15.3.2 The affine Galilei group (* 344
15.3.3 The (restricted) Schrodinger group W 347
15.3.4 The affine Poincare group (* 349
16 The Discretization Problem: Frames, Sampling,
and All That 353
16.1 The Weyl Heisenberg group or canonical CS 354
16.2 Wavelet frames 357
Contents xiii
16.3 Frames for affine semidirect products 359
16.3.1 The affine Weyl Heisenberg group 359
16.3.2 The affine Poincare groups (A) 360
16.3.3 Discrete frames for general semidirect products . 362
16.4 Groups without dilations: The Poincare groups (*) ... 364
16.5 A group theoretical approach to discrete
wavelet transforms 369
16.5.1 Generalities on sampling 369
16.5.2 Wavelets on 1V revisited * 370
16.5.3 Wavelets on a discrete abelian group 372
16.6 Conclusion 387
Conclusion and Outlook 389
References 393
Index 415
|
| any_adam_object | 1 |
| author | Ali, Syed Twareque Antoine, Jean-Pierre 1940- Gazeau, Jean-Pierre |
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| spelling | Ali, Syed Twareque Verfasser (DE-588)121807401 aut Coherent states, wavelets and their generalizations Syed Twareque Ali ; Jean-Pierre Antoine ; Jean-Pierre Gazeau New York [u.a.] Springer 2000 XIII, 418 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate texts in contemporary physics Etats cohérents ram Groepentheorie gtt Lie-groepen gtt Ondelettes ram Coherent states Wavelets (Mathematics) Quantenphysik (DE-588)4266670-3 gnd rswk-swf Wavelet (DE-588)4215427-3 gnd rswk-swf Kohärenter Zustand (DE-588)4125526-4 gnd rswk-swf Quantenphysik (DE-588)4266670-3 s Wavelet (DE-588)4215427-3 s DE-604 Kohärenter Zustand (DE-588)4125526-4 s 1\p DE-604 Antoine, Jean-Pierre 1940- Verfasser (DE-588)121807444 aut Gazeau, Jean-Pierre Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008886842&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Ali, Syed Twareque Antoine, Jean-Pierre 1940- Gazeau, Jean-Pierre Coherent states, wavelets and their generalizations Etats cohérents ram Groepentheorie gtt Lie-groepen gtt Ondelettes ram Coherent states Wavelets (Mathematics) Quantenphysik (DE-588)4266670-3 gnd Wavelet (DE-588)4215427-3 gnd Kohärenter Zustand (DE-588)4125526-4 gnd |
| subject_GND | (DE-588)4266670-3 (DE-588)4215427-3 (DE-588)4125526-4 |
| title | Coherent states, wavelets and their generalizations |
| title_auth | Coherent states, wavelets and their generalizations |
| title_exact_search | Coherent states, wavelets and their generalizations |
| title_full | Coherent states, wavelets and their generalizations Syed Twareque Ali ; Jean-Pierre Antoine ; Jean-Pierre Gazeau |
| title_fullStr | Coherent states, wavelets and their generalizations Syed Twareque Ali ; Jean-Pierre Antoine ; Jean-Pierre Gazeau |
| title_full_unstemmed | Coherent states, wavelets and their generalizations Syed Twareque Ali ; Jean-Pierre Antoine ; Jean-Pierre Gazeau |
| title_short | Coherent states, wavelets and their generalizations |
| title_sort | coherent states wavelets and their generalizations |
| topic | Etats cohérents ram Groepentheorie gtt Lie-groepen gtt Ondelettes ram Coherent states Wavelets (Mathematics) Quantenphysik (DE-588)4266670-3 gnd Wavelet (DE-588)4215427-3 gnd Kohärenter Zustand (DE-588)4125526-4 gnd |
| topic_facet | Etats cohérents Groepentheorie Lie-groepen Ondelettes Coherent states Wavelets (Mathematics) Quantenphysik Wavelet Kohärenter Zustand |
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