Ali / Antoine / Gazeau Coherent States, Wavelets and Their Generalizations

Erscheinungsjahr 2012
ISBN: 978-1-4612-1258-4
Verlag: Springer US
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 418 Seiten, eBook

Reihe: Graduate Texts in Contemporary Physics

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.

Zielgruppe

Research

Autoren/Hrsg.

Ali, Syed T.

Antoine, J-P

Gazeau, Jean-Perre

Inhaltsverzeichnis

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.- 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.- 4.3 An extended Schur’s lemma.- 4.4 Harmonic analysis on locally compact abelian groups.- 4.5 Lie groups and Lie algebras: A reminder.- 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.- 5.4 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.- 7 Covariant Coherent States.- 7.1 Covariant coherent states.- 7.2 Example: The classical theory of coherent states.- 7.3 Square integrable covariant CS: The general case.- 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.- 9.2 A generalization: ?- and V-admissibility.- 10 CS of General Semidirect Product Groups.- 10.1 Squeezed states (?).- 10.2 Geometry of semidirect product groups.- 10.3 CS of semidirect products.- 11 CS ofthe Relativity Groups.- 11.1 The Poincaré groups (1, 3) and (1,1).- 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.- 12.5 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.- 13.2 Towards a fast CWT: Continuous wavelet packets.- 13.3 Wavelets on the finite field ?p (?).- 13.4 Algebraic wavelets.- 14 Multidimensional Wavelets.- 14.1 Going to higher dimensions.- 14.2 Mathematical analysis (?).- 14.3 The 2-D case.- 15 Wavelets Related to Other Groups.- 15.1 Wavelets on the sphere and similar manifolds.- 15.2 The affine Weyl-Heisenberg group (?).- 15.3 The affine or similitude groups of space-time.- 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.- 16.4 Groups without dilations: The Poincaré groups (?).- 16.5 A group-theoretical approach to discrete wavelet transforms.- 16.6 Conclusion.- Conclusion and Outlook.- References.