# zbMATH — the first resource for mathematics

Coherent states, wavelets and their generalizations. (English) Zbl 1064.81069
Graduate Texts in Contemporary Physics. New York, NY: Springer (ISBN 0-387-98908-0/hbk). xiii, 418 p. (2000).
The subject of coherent states and/or wavelets (CS-W) is a hot topic for decades. There are many thousands of published papers, among them one can find about a hundred books and conference proceedings. Could one more book be a noticeable drop to the ocean? Personally I prefer the book under review to many other ones published before. It is common in the literature to treat wavelets separately from coherent states. Also wavelets are often considered without their connections to group representations and as a result these links are unknown to some active researchers and theory consumers (e.g. engineers). Consequently the general concept is lost and people are left with a disintegrated set of examples, tricks, and applications. The book under review consolidates the subject of CS-W putting it on the solid base of representation theory, systematically applying this approach to different kinds of CS-W and their numerous generalisations. The book could be considered as an extended version of the survey by S. Twareque Ali, J.-P. Antoine, J.-P. Gazeau and U.A. Mueller, Rev. Math. Phys. 7, No. 7, 1013–1104 (1995; Zbl 0837.43014)] of the same authors and A. U. Mueller.
The contents of the book is as follows: 1. Introduction; 2. Canonical coherent states; 3. Positive Operator-Valued Measures and Frames; 4. Some Group Theory; 5. Hilbert Spaces with Reproducing Kernels and Coherent States; 6. Square Integrable and Holomorphic Kernels; 7. Covariant Coherent States; 8. Coherent States from Square Integrable Representations; 9. Some Examples and Genralizations; 10. CS of General Semidirect Product Groups; 11. CS of Relativity Groups; 12. Wavelets; 13. Discrete Wavelet Transforms; 14. Multidimensional Wavelets; 15. Wavelets Related to Other Groups; 16. The Discretization Problem: Frames, Sampling, and All That.
The book contains all necessary facts from representation theory. It is clearly written, mathematically sound and well illustrated by diverse examples and applications. Probably due to the authors’ personal interests a larger part of the material is about coherent states, i.e. it is linked to physics and quantum mechanics. The important examples related to $$SU(1,1)$$, the Lorentz, the Galilei, and the Heisenberg(-Weyl) groups are considered in particular detail. Coverage of wavelets (i.e. motivated by signal processing) is also sufficient and given from the same unifying viewpoint of representation theory of the $$ax+b$$ (affine) group.
It is impossible and useless to compile “the complete list of references” on the subject of CS-W. 290 references given in the book form a good attempt to present a list sufficient for most purposes and yet practical. Short but informative historical remarks in the text additionally guide through the literature. The index is useful too.
As a small criticism of the book under review I would like to mention the absence of (formal) exercises which one normally expects from a book in a graduate text series. Besides pedagogical needs well chosen exercises allow to extend the scope of consideration without a significant increment of the book length.
Another comment is more related to the area of CS-W in general than to the particular book. CS-W are not only an applied tool in quantum mechanics and signal processing. The same construction in facts describes most of analytic function theories. For example CS-W are useful in the study of Toeplitz operators. This “pure” side of CS-W should gain an equal attention to applications in literature, especially in textbooks.
I recommend this book to everyone who wishes to learn CS-W, or already works in the area, or just needs a good reference source.

##### MSC:
 81R30 Coherent states 22E70 Applications of Lie groups to the sciences; explicit representations 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems 81-02 Research exposition (monographs, survey articles) pertaining to quantum theory