Ebook: Coherent States, Wavelets and Their Generalizations

This book presents a survey of the theory of coherent states, wavelets, and some of their generalizations, emphasizing mathematical structures. The point of view is that both the theories of both wavelets and coherent states can be subsumed into a single analytic structure. 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 otherwise obscure properties of wavelets and of coherent states. Many concrete examples, such as semisimple Lie groups, the relativity group, and several kinds of wavelets, are discussed in detail. The book concludes with physical applications, centering on the quantum measurement problem and the quantum-classical transition. 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 book will also be a useful compendium of recent results for physicists and mathematicians already active in the field.

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This book presents a survey of the theory of coherent states, wavelets, and some of their generalizations, emphasizing mathematical structures. The point of view is that both the theories of both wavelets and coherent states can be subsumed into a single analytic structure. 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 otherwise obscure properties of wavelets and of coherent states. Many concrete examples, such as semisimple Lie groups, the relativity group, and several kinds of wavelets, are discussed in detail. The book concludes with physical applications, centering on the quantum measurement problem and the quantum-classical transition. 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 book will also be a useful compendium of recent results for physicists and mathematicians already active in the field.
Content:
Front Matter....Pages i-xiii
Introduction....Pages 1-12
Canonical Coherent States....Pages 13-32
Positive Operator-Valued Measures and Frames....Pages 33-45
Some Group Theory....Pages 47-87
Hilbert Spaces with Reproducing Kernels and Coherent States....Pages 89-108
Square Integrable and Holomorphic Kernels....Pages 109-125
Covariant Coherent States....Pages 127-145
Coherent States from Square Integrable Representations....Pages 147-170
Some Examples and Generalizations....Pages 171-197
CS of General Semidirect Product Groups....Pages 199-223
CS of the Relativity Groups....Pages 225-256
Wavelets....Pages 257-282
Discrete Wavelet Transforms....Pages 283-305
Multidimensional Wavelets....Pages 307-330
Wavelets Related to Other Groups....Pages 331-351
The Discretization Problem: Frames, Sampling, and All That....Pages 353-387
Conclusion and Outlook....Pages 389-391
Back Matter....Pages 393-418

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This book presents a survey of the theory of coherent states, wavelets, and some of their generalizations, emphasizing mathematical structures. The point of view is that both the theories of both wavelets and coherent states can be subsumed into a single analytic structure. 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 otherwise obscure properties of wavelets and of coherent states. Many concrete examples, such as semisimple Lie groups, the relativity group, and several kinds of wavelets, are discussed in detail. The book concludes with physical applications, centering on the quantum measurement problem and the quantum-classical transition. 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 book will also be a useful compendium of recent results for physicists and mathematicians already active in the field.
Content:
Front Matter....Pages i-xiii
Introduction....Pages 1-12
Canonical Coherent States....Pages 13-32
Positive Operator-Valued Measures and Frames....Pages 33-45
Some Group Theory....Pages 47-87
Hilbert Spaces with Reproducing Kernels and Coherent States....Pages 89-108
Square Integrable and Holomorphic Kernels....Pages 109-125
Covariant Coherent States....Pages 127-145
Coherent States from Square Integrable Representations....Pages 147-170
Some Examples and Generalizations....Pages 171-197
CS of General Semidirect Product Groups....Pages 199-223
CS of the Relativity Groups....Pages 225-256
Wavelets....Pages 257-282
Discrete Wavelet Transforms....Pages 283-305
Multidimensional Wavelets....Pages 307-330
Wavelets Related to Other Groups....Pages 331-351
The Discretization Problem: Frames, Sampling, and All That....Pages 353-387
Conclusion and Outlook....Pages 389-391
Back Matter....Pages 393-418
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