Coherent States, Wavelets and Their Generalizations

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.