Wavelets have become an important analytical tool in signal processing and other applications; coherent states provide a useful analytic framework in many areas of physics, including quantum optics and atomic physics. This book presents a unified survey of the theories of coherent states, wavelets, and some of their generalizations, emphasizing mathematical structures. Many concrete examples are discussed in detail. 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

Klappentext

A survey of the theory of coherent states, wavelets, and some of their generalizations, emphasizing mathematical structures. 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 the group context to be dispensed with altogether. 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.

Inhalt

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.