Coherent States, Wavelets, and Their Generalizations

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Springer Science & Business Media, Oct 30, 2013 - Science - 577 pages

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 research literature, 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.

 

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Contents

Chapter 1 Introduction
1
Chapter 2 Canonical Coherent States
15
Chapter 3 Positive OperatorValued Measures and Frames
37
Chapter 4 Some Group Theory
61
Chapter 5 Hilbert Spaces with Reproducing Kernels and Coherent States
104
Chapter 6 Square Integrable and Holomorphic Kernels
133
Chapter 7 Covariant Coherent States
165
Chapter 8 Coherent States from Square Integrable Representations
203
Chapter 12 Wavelets
347
Chapter 13 Discrete Wavelet Transforms
378
Chapter 14 Multidimensional Wavelets and Generalizations
411
Chapter 15 Wavelets on Manifolds
457
Chapter 16 Wavelets Related to Affine Groups
495
Frames Sampling and All That
514
Chapter 18 Conclusion and Outlook
537
References
541

Chapter 9 CS of General Semidirect Product Groups
245
Chapter 10 CS of the Relativity Groups
270
Chapter 11 Integral Quantization
305

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About the author (2013)

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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