By Syed Twareque Ali, Jean-Pierre Antoine, Jean-Pierre Gazeau

ISBN-10: 1461212588

ISBN-13: 9781461212584

ISBN-10: 1461270650

ISBN-13: 9781461270652

This ebook provides a survey of the idea of coherent states, wavelets, and a few in their generalizations, emphasizing mathematical buildings. the viewpoint is that either the theories of either wavelets and coherent states may be subsumed right into a unmarried analytic constitution. ranging from the normal concept of coherent states over Lie teams, the authors generalize the formalism via associating coherent states to team representations which are sq. integrable over a homogeneous area; an extra step permits one to dispense with the gang context altogether. during this context, wavelets might be generated from coherent states of the affine team of the genuine line, and higher-dimensional wavelets come up from coherent states of different teams. The unified history makes obvious differently vague homes of wavelets and of coherent states. Many concrete examples, comparable to semisimple Lie teams, the relativity workforce, and several other sorts of wavelets, are mentioned intimately. The publication concludes with actual purposes, centering at the quantum dimension challenge and the quantum-classical transition. meant as an advent to present learn for graduate scholars and others coming into the sphere, the mathematical dialogue is self- contained. With its huge references to the examine literature, the e-book can also be an invaluable compendium of modern effects for physicists and mathematicians already lively within the box.

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**Extra info for Coherent States, Wavelets and Their Generalizations**

**Example text**

All group spaces and parameter spaces for defining CS will be of this type. ) The Borel sets 8(X) of X consist of elements of the O'-algebra formed by its open sets. + U {oo} satisfying the following properties. 1. 1) IJ(0) = O. 2. 3) iEJ If, in addition, IJ satisfies sup IJ(~) = O~A, 0 = compact then it is called a regular Borel measure. Unless the contrary is stated, all Borel measures will be assumed to be regular. The measure IJ is said to be bounded or finite if IJ(X) < 00. While referring to a measure IJ, we shall use either of the two notations, IJ or dIJ.

Canonical Coherent States Hilbert spaces. This will be dealt with in Chapter 5, and it will also take us into the theory of frames in Chapters 7 and 16. ) To bring out some complex analytic properties of the canonical CS, let ¢ E iJ be an arbitrary vector. ] (nl¢) zn 2 n=O v'n1 (zl¢) exp [J~2] J(z). 60) Here J is an analytic function of the complex variable z. 35) as dv(q,p) = dZAdZ 2. 61) 7rZ and let us define the new measure _ 2 dJL(z, z) = exp [-Izl] dz A dZ 2. 62) . In this notation, "A" denotes the exterior product of the two differentials dz and az (considered as one-forms on the complex manifold C).

L¢ above is bounded. In particular, the POV measure is said to be normalized if a(X) = I (= identity operator on S)). , for each x E X, F(x) is a bounded, positive operator, and, for arbitrary ¢,1jJ E S), the complex function x I-t (¢IF(x)1jJ) is measurable in B(X) [that is, the inverse image of any Borel subset of C belongs to B(X)]. Using F and the regular Borel measure v, we may define the regular POV measure a, such that for all ~ E B(X), a(~) = i F(x) dv(x). 7) The operator function F is then called a density for the POV measure a.

### Coherent States, Wavelets and Their Generalizations by Syed Twareque Ali, Jean-Pierre Antoine, Jean-Pierre Gazeau

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