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E-grāmata: Coherent States, Wavelets, and Their Generalizations

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Coherent States, Wavelets, and Their GeneralizationsPalielināt
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This second edition is fully updated, covering in particular new types of coherent states (the so-called Gazeau-Klauder coherent states, nonlinear coherent states, squeezed states, as used now routinely in quantum optics) and various generalizations of wavelets (wavelets on manifolds, curvelets, shearlets, etc.). In addition, it contains a new chapter on coherent state quantization and the related probabilistic aspects. As a survey of the theory of coherent states, wavelets, and some of their generalizations, it emphasizes mathematical principles, subsuming the theories of both wavelets and coherent states into a single analytic structure. The approach allows the user to take a classical-like view of quantum states in physics.

Starting from the standard theory of coherent states over Lie groups, the authors generalize the formalism by associating coherent states to group representations that are square integrable over a homogeneous space; a further step allows one to dispense with the group context altogether. In this context, wavelets can be generated from coherent states of the affine group of the real line, and higher-dimensional wavelets arise from coherent states of other groups. The unified background makes transparent an entire range of properties of wavelets and coherent states. Many concrete examples, such as coherent states from semisimple Lie groups, Gazeau-Klauder coherent states, coherent states for the relativity groups, and several kinds of wavelets, are discussed in detail. The book concludes with a palette of potential applications, from the quantum physically oriented, like the quantum-classical transition or the construction of adequate states in quantum information, to the most innovative techniques to be used in data processing.

Intended as an introduction to current research for graduate students and others entering the field, the mathematical discussion is self-contained. With its extensive references to the researchliterature, the first edition of the book is already a proven compendium for physicists and mathematicians active in the field, and with full coverage of the latest theory and results the revised second edition is even more valuable.
1 Introduction
1(14)
2 Canonical Coherent States
15(22)
2.1 Minimal Uncertainty States
16(4)
2.2 The Group Theoretical Backdrop
20(3)
2.3 Some Functional Analytic Properties
23(4)
2.4 A Complex Analytic Viewpoint
27(4)
2.5 An Alternative Representation and Squeezed States
31(3)
2.6 Some Geometrical Considerations
34(1)
2.7 Outlook
35(2)
3 Positive Operator-Valued Measures and Frames
37(24)
3.1 Definitions and Main Properties
38(5)
3.1.1 Examples of POV-Measures
41(2)
3.2 The Case of a Tight Frame
43(2)
3.3 Frames and Semi-frames
45(9)
3.3.1 Frames Revisited
45(1)
3.3.2 Upper Semi-frames
46(3)
3.3.3 Lower Semi-frames, Duality
49(5)
3.4 Discrete Frames and Semi-frames
54(7)
3.4.1 Discrete Frames
54(3)
3.4.2 Weighted and Controlled Frames
57(1)
3.4.3 Fusion Frames
57(1)
3.4.4 Discrete Semi-frames
58(2)
3.4.5 Discretization
60(1)
4 Some Group Theory
61(44)
4.1 Homogeneous Spaces, Quasi-Invariant, and Invariant Measures
61(8)
4.1.1 A Simple Example
63(4)
4.1.2 An Example Using the Affine Group
67(2)
4.2 Induced Representations and Systems of Covariance
69(15)
4.2.1 Vector Coherent States
73(3)
4.2.2 Discrete Series Representations of SU(1,1)
76(7)
4.2.3 The Regular Representations of a Group
83(1)
4.3 An Extended Schur's Lemma
84(2)
4.4 Harmonic Analysis on Locally Compact Abelian Groups
86(4)
4.4.1 Basic Notions
86(2)
4.4.2 Lattices in LCA Groups
88(1)
4.4.3 Sampling in LCA Groups
89(1)
4.5 Lie Groups and Lie Algebras: A Reminder
90(15)
4.5.1 Lie Algebras
91(2)
4.5.2 Lie Groups
93(5)
4.5.3 Extensions of Lie Algebras and Lie Groups
98(3)
4.5.4 Contraction of Lie Algebras and Lie Groups
101(4)
5 Hilbert Spaces with Reproducing Kernels and Coherent States
105(28)
5.1 A First Look at Reproducing Kernels
106(5)
5.2 Some Illustrative Examples
111(3)
5.2.1 The Canonical CS
111(1)
5.2.2 An Example from a Hardy Space
111(1)
5.2.3 An Example of VCS from a Matrix Domain
112(2)
5.3 A Second Look at Reproducing Kernels
114(3)
5.3.1 A Motivating Example
114(1)
5.3.2 Measurable Fields and Direct Integrals
115(2)
5.3.3 Example Using a POV Function
117(1)
5.4 Reproducing Kernel Hilbert Spaces: General Construction
117(13)
5.4.1 Positive-Definite Kernels and Evaluation Maps
118(4)
5.4.2 Coherent States and POV Functions
122(3)
5.4.3 Some Isomorphisms, Bases, and v-Selections
125(3)
5.4.4 A Reconstruction Problem: Example of a Holomorphic Map
128(2)
5.5 Some Properties of Reproducing Kernel Hilbert Spaces
130(3)
6 Square Integrable and Holomorphic Kernels
133(32)
6.1 Square Integrable Kernels
134(2)
6.2 Holomorphic Kernels
136(4)
6.3 Some Examples of Coherent States from Square Integrable Kernels
140(11)
6.3.1 Standard Versus Circle Coherent States
141(1)
6.3.2 Coherent States for Motion on the Circle
141(2)
6.3.3 A General Holomorphic Construction
143(3)
6.3.4 Nonlinear Coherent States and Orthogonal Polynomials
146(5)
6.4 Gazeau--Klauder CS
151(12)
6.4.1 Coherent States for Discrete Spectrum Dynamics
151(2)
6.4.2 Statistical and Semi-Classical Aspects
153(2)
6.4.3 Imposing the Hamiltonian Lower Symbol
155(1)
6.4.4 Action-Angle Coherent States
156(1)
6.4.5 Two Examples of Action-Angle Coherent States
157(3)
6.4.6 Coherent States for Continuum Dynamics
160(1)
6.4.7 Coherent States for Discrete and Continuum Dynamics
161(1)
6.4.8 A General Expression
162(1)
6.5 CS on Quaternionic Hilbert Spaces and Hilbert Modules
163(2)
7 Covariant Coherent States
165(38)
7.1 Square-Integrable Covariant Coherent States
166(8)
7.1.1 A General Definition
166(2)
7.1.2 The Gilmore--Perelomov CS
168(2)
7.1.3 Vector and Matrix CS: A Geometrical Setting
170(4)
7.2 Example: The Classical Theory of Coherent States
174(6)
7.2.1 CS of Compact Semisimple Lie Groups
174(3)
7.2.2 CS of Noncompact Semisimple Lie Groups
177(2)
7.2.3 CS of Non-Semisimple Lie Groups
179(1)
7.3 Covariant CS: The General Case
180(6)
7.3.1 Generalized Perelomov CS
182(1)
7.3.2 Continuous Semi-Frames
183(1)
7.3.3 Some Further Generalizations
184(2)
7.4 CS on Spheres Through Heat Kernels
186(7)
7.5 CS in Loop Quantum Gravity and Quantum Cosmology
193(1)
7.6 CS on Conformal Classical Domains
194(9)
7.6.1 Beyond Square Integrability and Singular Orbit
201(2)
8 Coherent States from Square Integrable Representations
203(42)
8.1 Square Integrable Group Representations
204(8)
8.1.1 Example: The Connected Affine Group
209(3)
8.2 Orthogonality Relations
212(4)
8.3 A Class of Semidirect Product Groups
216(10)
8.3.1 Three Concrete Examples
221(4)
8.3.2 A Broader Setting
225(1)
8.4 A Generalization: α and V-Admissibility
226(19)
8.4.1 Example of the Galilei Group
231(5)
8.4.2 CS of the Isochronous Galilei Group
236(7)
8.4.3 Atomic Coherent States
243(2)
9 CS of General Semidirect Product Groups
245(26)
9.1 Squeezed States
246(4)
9.2 Geometry of Semidirect Product Groups
250(10)
9.2.1 A Special Class of Orbits
250(2)
9.2.2 The Coadjoint Orbit Structure of Γ
252(4)
9.2.3 Measures on Γ
256(2)
9.2.4 Induced Representations of Semidirect Products
258(2)
9.3 CS of Semidirect Products
260(11)
9.3.1 Admissible Affine Sections
267(4)
10 CS of the Relativity Groups
271(34)
10.1 The Poincare Groups p+↑(1, 3) and p+↑(1, 1)
271(19)
10.1.1 The Poincare Group p+↑(1, 3) in 1 + 3 Dimensions
271(11)
10.1.2 The Poincare Group p+↑(1, 1) in 1 + 1 Dimensions
282(5)
10.1.3 Poincare CS: The Massless Case
287(3)
10.2 The Galilei Groups g(1, 1) and g ≡ g(3, 1)
290(2)
10.3 The (1 + 1)-Dimensional Anti-de Sitter Group SO°(1, 2) and Its Contraction(s)
292(13)
11 Integral Quantization
305(42)
11.1 What is Really Quantization?
306(1)
11.2 Sea Star Algebra
307(6)
11.2.1 Exploring the Plane with a Five-Fold Set of Arms
307(4)
11.2.2 What About the Continuous Frame?
311(1)
11.2.3 Probabilistic Aspects
312(1)
11.2.4 Beyond Coherent States: Integral Quantization
312(1)
11.3 Integral Quantization
313(3)
11.3.1 Covariant Integral Quantization
314(2)
11.4 Exemple: Weyl-Heisenberg Covariant Integral Quantization(s)
316(7)
11.4.1 Quantization with a Generic Weight Function
316(2)
11.4.2 Regular and Isometric Quantizations
318(1)
11.4.3 Elliptic Regular Quantizations
319(2)
11.4.4 Elliptic Regular Quantizations that Are Isometric
321(1)
11.4.5 Hyperbolic Regular Quantizations
321(1)
11.4.6 Hyperbolic Regular Quantizations that Are Isometric
321(1)
11.4.7 Quantum Harmonic Oscillator According to ω
321(1)
11.4.8 Variations on the Wigner Function
322(1)
11.5 Weyl-Heisenberg Integral Quantizations of Functions and Distributions
323(9)
11.5.1 Acceptable Probes ρ
323(2)
11.5.2 Quantizable Functions
325(3)
11.5.3 Quantizable Distributions
328(4)
11.6 Quantization with Coherent States or Frame: General
332(12)
11.6.1 A First Example: Frame Quantization of Finite Sets
333(6)
11.6.2 A Second Example: CS Quantization of Motion on the Circle
339(3)
11.6.3 Quantization With Action-Angle CS for Bounded Motions
342(2)
11.7 Application to Various Systems
344(3)
12 Wavelets
347(32)
12.1 A Word of Motivation
347(4)
12.2 Derivation and Properties of the 1-D Continuous Wavelet Transform
351(6)
12.3 A Mathematical Aside: Extension to Distributions
357(5)
12.4 Interpretation of the Continuous Wavelet Transform
362(3)
12.4.1 The CWT As Phase Space Representation
362(1)
12.4.2 Localization Properties and Physical Interpretation of the CWT
363(2)
12.5 Discretization of the Continuous WT: Discrete Frames
365(2)
12.6 Ridges and Skeletons
367(1)
12.7 Applications
368(11)
12.7.1 Application to NMR Spectroscopy
370(9)
13 Discrete Wavelet Transforms
379(32)
13.1 The Discrete WT
379(6)
13.1.1 Multiresolution Analysis, Orthonormal Wavelet Bases
380(2)
13.1.2 Connection with Filters and the Subband Coding Scheme
382(2)
13.1.3 Generalizations
384(1)
13.1.4 Applications
385(1)
13.2 Towards a Fast CWT: Continuous Wavelet Packets
385(2)
13.3 Algebraic Wavelets
387(3)
13.4 A Group-Theoretical Approach to Discrete Wavelet Transforms
390(4)
13.4.1 Wavelets on the Finite Field Zp
390(2)
13.4.2 Wavelets on Zp: Pseudodilations and Group Structure
392(2)
13.5 Wavelets on a Discrete Abelian Group
394(17)
13.5.1 Compatible Filters: The General Case
395(5)
13.5.2 Compatible Filters in the Case G = R, A ⊂ Z+
400(2)
13.5.3 Cohomological Interpretation
402(2)
13.5.4 Compatible Filters and Discretized Wavelet Transform
404(7)
14 Multidimensional Wavelets and Generalizations
411(46)
14.1 Going to Higher Dimensions
411(1)
14.2 Mathematical Analysis
412(11)
14.2.1 The CWT in n Dimensions
412(9)
14.2.2 The CWT of Radial Functions
421(2)
14.3 The Two-Dimensional CWT
423(11)
14.3.1 Minimality Properties
423(3)
14.3.2 Interpretation, Visualization Problems, Calibration
426(3)
14.3.3 Practical Applications of the CWT in Two Dimensions
429(5)
14.4 The Discrete WT in Higher Dimensions
434(4)
14.4.1 Applications of the 2-D DWT
436(1)
14.4.2 The DWT of Radial Functions
437(1)
14.5 Generalizations of 2-D Wavelets
438(19)
14.5.1 Continuous Wavelet Packets in Two Dimensions
439(1)
14.5.2 General Directional Wavelets
440(1)
14.5.3 Multiselective Wavelets
440(5)
14.5.4 Ridgelets
445(2)
14.5.5 Curvelets and Contourlets
447(3)
14.5.6 Shearlets
450(5)
14.5.7 Geometrical "Wavelets": Dictionaries, Molecules
455(2)
15 Wavelets on Manifolds
457(38)
15.1 Wavelets on the Two-Sphere
457(20)
15.1.1 Stereographic Wavelets on the Two-Sphere
458(12)
15.1.2 Poisson Wavelets on the Sphere
470(1)
15.1.3 Discrete Spherical Wavelets
471(6)
15.2 Wavelets on Other Manifolds
477(14)
15.2.1 Wavelets on Conic Sections
477(2)
15.2.2 Wavelets on the n-Sphere and the Two-Torus
479(3)
15.2.3 Wavelets on Surfaces of Revolution
482(6)
15.2.4 Local Wavelets
488(3)
15.3 Wavelets on Graphs
491(4)
16 Wavelets Related to Affine Groups
495(20)
16.1 The Affine Weyl--Heisenberg Group
495(4)
16.2 The Affine or Similitude Groups of Spacetime
499(11)
16.2.1 Kinematical Wavelets, Motion Analysis
499(3)
16.2.2 The Affine Galilei Group
502(4)
16.2.3 The (Restricted) Galilei--Schrodinger Group
506(3)
16.2.4 The Affine Poincare Group and the Conformal Group
509(1)
16.3 Some Generalizations: Wavelets on Riemannian Symmetric Spaces
510(5)
17 The Discretization Problem: Frames, Sampling, and All That
515(22)
17.1 The Weyl--Heisenberg Group or Canonical CS
516(3)
17.2 Wavelet Frames
519(2)
17.2.1 Alternative Approaches to the "ax + b" Group
520(1)
17.3 Frames for Affine Semidirect Products
521(5)
17.3.1 The Affine Weyl--Heisenberg Group
521(1)
17.3.2 The Affine Poincare Groups
521(3)
17.3.3 Discrete Frames for General Semidirect Products
524(2)
17.4 Groups Without Dilations: The Poincare Groups
526(10)
17.4.1 The Poincare Group p+↑ (1, 1)
527(5)
17.4.2 The Poincare Group p+↑ (1, 3)
532(4)
17.5 Generalities on Sampling
536(1)
18 Conclusion and Outlook
537(4)
18.1 Present Status of CS and Wavelet Research
537(2)
18.2 What Is the Future from Our Point of View?
539(2)
References 541(32)
Index 573
Dr. S. T. Ali is a full professor in the Department of Mathematics and Statistics, Concordia University, Montreal. He has held teaching and research positions at the Abdus Salam International Centre for Theoretical Physics (Trieste, Italy); the Department of Mathematics (University of Toronto, Canada); the Institut fur Theoretische Physike (Technische Universitat Clausthal, Germany); He has written about 150 research publications, including 3 books.

JP. Antoine has a full career as a professor of mathematical physics at Universite Catholique de Louvain, Belgium, after postdocs at Princeton University, University of Pittsburgh, and University de Geneve. He was also invited professor at many foreign universities (Universite Paris 7, Universite Burundi, Fukuoka University (Japan), Concordia University, Universite Nationale du Benin). He is now professor emeritus, but continues his research activities. At the Universite Catholique de Louvain, he was head of the Institute of Theoretical and Mathematical Physics (FYMA), Chairman of the Department of Physics, and member of the European Physical Society, the American Physical Society, the International Association of Mathematical Physics, the American Mathematical Society. He is a referee for a dozen of international journals of physics, mathematical physics, and signal processing. His research interests cover a wide range. From mathematical physics, in particular, the formulation of quantum mechanics beyond Hilbert space, he moved to several innovative topics in pure mathematics, such as partial inner product spaces and partial algebras of unbounded operators, a framework in which he is still working actively. On the other hand, he also pursued several research directions in mathematical physics, namely, classical gauge field theories, coherent states and wavelets, including the applications of the latter in signal and image processing. He is the author or co-author of more than 120 research papers, four monographs and plenty of book chapters or conference papers.

J-P. Gazeau is a full professor of Physics at the University Paris Diderot (Sorbonne Paris Cite), France, a member of the "Astroparticles and Cosmology" Laboratory (CNRS, UMR 7164), and currently Chairman of the Standing Committee of the International Colloquium on Group Theoretical Methods in Physics. Having obtained his academic degrees from Sorbonne University and Pierre-and Marie Curie University (Paris 6), he spent most of his academic career in Paris and as invited professor and researcher, in many other places, among them UCLA, Louvain, Montreal, Prague, Newcastle, Krakow, Rio de Janeiro and Sao Paulo. Professor Gazeau has authored more than 160 scientific publications, including 2 books, in theoretical and Mathematical Physics, mostly devoted to group theoretical methods in physics, coherent states, quantization methods, and number theory for aperiodic systems.
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