## Coherent States, Wavelets, and Their GeneralizationsThis 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

1 | |

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 |

### Other editions - View all

Coherent States, Wavelets and Their Generalizations Syed T. Ali,J-P Antoine,Jean-Perre Gazeau Limited preview - 2012 |

Coherent States, Wavelets, and Their Generalizations Syed Twareque Ali,Jean-Pierre ANTOINE,JP Gazeau No preview available - 2016 |