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Physics - Quantum Physics | Coherent States, Wavelets and Their Generalizations

Coherent States, Wavelets and Their Generalizations

Ali, Syed T., ANTOINE, Jean-Pierre, Gazeau, Jean-Perre

2000, XIII, 418 p.

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Nitya kaaler utshab taba Bishyer-i-dipaalika Aami shudhu tar-i-mateer pradeep Jaalao tahaar shikhaa 1 - Tagore Should authors feel compelled to justify the writing of yet another book? In an overpopulated world, should parents feel compelled to justify bringing forth yet another child? Perhaps not! But an act of creation is also an act of love, and a love story can always be happily shared. In writing this book, it has been our feeling that, in all of the wealth of material on coherent states and wavelets, there exists a lack of a discern­ able, unifying mathematical perspective. The use of wavelets in research and technology has witnessed explosive growth in recent years, while the use of coherent states in numerous areas of theoretical and experimental physics has been an established trend for decades. Yet it is not at all un­ common to find practitioners in either one of the two disciplines who are hardly aware of one discipline's links to the other. Currently, many books are on the market that treat the subject of wavelets from a wide range of perspectives and with windows on one or several areas of a large spectrum IThine is an eternal celebration '" A cosmic Festival of Lights! '" Therein I am a mere flicker of a wicker lamp, , . 0 kindle its flame (my Master!), vi Preface of possible applications.

Content Level » Research

Keywords » Coherent states - Lie groups - group theory - harmonic analysis - quantum measurement problem - quantum-classical transition - relativity groups - wavelets

Related subjects » Algebra - Applied & Technical Physics - Quantum Physics

Table of contents 

1 Introduction.- 2 Canonical Coherent States.- 2.1 Minimal uncertainty states.- 2.2 The group-theoretical backdrop.- 2.3 Some functional analytic properties.- 2.4 A complex analytic viewpoint (?).- 2.5 Some geometrical considerations.- 2.6 Outlook.- 2.7 Two illustrative examples.- 2.7.1 A quantization problem (?).- 2.7.2 An application to atomic physics (?).- 3 Positive Operator-Valued Measures and Frames.- 3.1 Definitions and main properties.- 3.2 The case of a tight frame.- 3.3 Example: A commutative POV measure.- 3.4 Discrete frames.- 4 Some Group Theory.- 4.1 Homogeneous spaces, quasi-invariant, and invariant measures.- 4.2 Induced representations and systems of covariance.- 4.2.1 Vector coherent states.- 4.2.2 Discrete series representations of SU(1,1).- 4.2.3 The regular representations of a group.- 4.3 An extended Schur’s lemma.- 4.4 Harmonic analysis on locally compact abelian groups.- 4.4.1 Basic notions.- 4.4.2 Lattices in LCA groups.- 4.4.3 Sampling in LCA groups.- 4.5 Lie groups and Lie algebras: A reminder.- 4.5.1 Lie algebras.- 4.5.2 Lie groups.- 4.5.3 Extensions of Lie algebras and Lie groups.- 4.5.4 Contraction of Lie algebras and Lie groups.- 5 Hilbert Spaces with Reproducing Kernels and Coherent States.- 5.1 A motivating example.- 5.2 Measurable fields and direct integrals.- 5.3 Reproducing kernel Hilbert spaces.- 5.3.1 Positive-definite kernels and evaluation maps.- 5.3.2 Coherent states and POV functions.- 5.3.3 Some isomorphisms, bases, and v-selections.- 5.3.4 A reconstruction problem: Example of a holomorphic map (?).- 5.4 Some properties of reproducing kernel Hilbert spaces.- 6 Square Integrable and Holomorphic Kernels.- 6.1 Square integrable kernels.- 6.2 Holomorphic kernels.- 6.3 Coherent states: The holomorphic case.- 6.3.1 An example of a holomorphic frame (?).- 6.3.2 A nonholomorphic excursion (?).- 7 Covariant Coherent States.- 7.1 Covariant coherent states.- 7.1.1 A general definition.- 7.1.2 The Gilmore-Perelomov CS and vector CS.- 7.1.3 A geometrical setting.- 7.2 Example: The classical theory of coherent states.- 7.2.1 CS of compact semisimple Lie groups.- 7.2.2 CS of noncompact semisimple Lie groups.- 7.2.3 CS of non-semisimple Lie groups.- 7.3 Square integrable covariant CS: The general case.- 7.3.1 Some further generalizations.- 8 Coherent States from Square Integrable Representations.- 8.1 Square integrable group representations.- 8.2 Orthogonality relations.- 8.3 The Wigner map.- 8.4 Modular structures and statistical mechanics.- 9 Some Examples and Generalizations.- 9.1 A class of semidirect product groups.- 9.1.1 Three concrete examples (?).- 9.1.2 A broader setting.- 9.2 A generalization: ?- and V-admissibility.- 9.2.1 Example of the Galilei group (?).- 9.2.2 CS of the isochronous Galilei group (?).- 9.2.3 Atomic coherent states.- 10 CS of General Semidirect Product Groups.- 10.1 Squeezed states (?).- 10.2 Geometry of semidirect product groups.- 10.2.1 A special class of orbits.- 10.2.2 The coadjoint orbit structure of ?.- 10.2.3 Measures on ?.- 10.2.4 Induced representations of semidirect products.- 10.3 CS of semidirect products.- 10.3.1 Admissible affine sections.- 11 CS of the Relativity Groups.- 11.1 The Poincaré groups (1, 3) and (1,1).- 11.1.1 The Poincaré group in 1+3 dimensions, (1, 3) (?).- 11.1.2 The Poincaré group in 1+1 dimensions, (1,1).- 11.1.3 Poincaré CS: The massless case.- 11.2 The Galilei groups (11) and (1, 3) (?).- 11.3 The anti-de Sitter group SOo(1,2) and its contraction(s) (?).- 12 Wavelets.- 12.1 A word of motivation.- 12.2 Derivation and properties of the 1-D continuous wavelet transform (?).- 12.3 A mathematical aside: Extension to distributions.- 12.4 Interpretation of the continuous wavelet transform.- 12.4.1 The CWT as phase space representation.- 12.4.2 Localization properties and physical interpretation of the CWT.- 12.5 Discretization of the continuous WT: Discrete frames.- 12.6 Ridges and skeletons.- 12.7 Applications.- 13 Discrete Wavelet Transforms.- 13.1 The discrete time or dyadic WT.- 13.1.1 Multiresolution analysis and orthonormal wavelet bases.- 13.1.2 Connection with filters and the subband coding scheme.- 13.1.3 Generalizations.- 13.1.4 Applications.- 13.2 Towards a fast CWT: Continuous wavelet packets.- 13.3 Wavelets on the finite field ?p (?).- 13.4 Algebraic wavelets.- 13.4.1 ?-wavelets of Haar on the line (?).- 13.4.2 Pisot wavelets, etc.(?).- 14 Multidimensional Wavelets.- 14.1 Going to higher dimensions.- 14.2 Mathematical analysis (?).- 14.3 The 2-D case.- 14.3.1 Minimality properties.- 14.3.2 Interpretation, visualization problems, and calibration.- 14.3.3 Practical applications of the CWT in two dimensions.- 14.3.4 The discrete WT in two dimensions.- 14.3.5 Continuous wavelet packets in two dimensions.- 15 Wavelets Related to Other Groups.- 15.1 Wavelets on the sphere and similar manifolds.- 15.1.1 The two-sphere (?).- 15.1.2 Generalization to other manifolds.- 15.2 The affine Weyl-Heisenberg group (?).- 15.3 The affine or similitude groups of space-time.- 15.3.1 Kinematical wavelets (?).- 15.3.2 The affine Galilei group (?).- 15.3.3 The (restricted) Schrödinger group (?).- 15.3.4 The affine Poincaré group (?).- 16 The Discretization Problem: Frames, Sampling, and All That.- 16.1 The Weyl—Heisenberg group or canonical CS.- 16.2 Wavelet frames.- 16.3 Frames for affine semidirect products.- 16.3.1 The affine Weyl-Heisenberg group.- 16.3.2 The affine Poincaré groups (?).- 16.3.3 Discrete frames for general semidirect products.- 16.4 Groups without dilations: The Poincaré groups (?).- 16.5 A group-theoretical approach to discrete wavelet transforms.- 16.5.1 Generalities on sampling.- 16.5.2 Wavelets on ?p revisited (?).- 16.5.3 Wavelets on a discrete abelian group.- 16.6 Conclusion.- Conclusion and Outlook.- References.

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