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**Generalized wavelets and hypergroups.**
*(English)*
Zbl 0926.42016

Amsterdam: Gordon & Breach. xii, 354 p. (1997).

The notion of hypergroup dates back to J. Delsarte, B. M. Levitan and is closely connected with ideas of I. M. Gelfand. Roughly speaking, a hypergroup is a pair \((X,*)\) where \(X\) is a locally compact space and “\(*\)” designates a convolution operator on the space of finite Borel measures on \(X\). A rigorous definition of the convolution “\(*\)” is based on the notion of a generalized translation operator. Harmonic analysis on hypergroups represents an interesting and important area in modern mathematics. It has applications to PDE, probability theory, integral geometry, physics etc.

The author is an expert in this area and an initiator of studying wavelet transforms on hypergroups. The book consists of ten chapters: I. Product formulas and generalized hypergroups. 2. Hypergroups. 3. Wavelets and the windowed spherical Fourier transform on Gelfand pairs. 4. Generalized wavelets and generalized continuous wavelet transforms on hypergroups. 5. Generalized wavelets and generalized continuous wavelet transforms on semisimple Lie groups and on Cartan motion groups. 6. Harmonic analysis, generalized wavelets and the generalized continuous wavelet transforms on Chébli-Trimèche hypergroups. 7. Harmonic analysis, generalized wavelets and the generalized continuous wavelet transform associated with the spherical mean operator. 8. Harmonic analysis, generalized wavelets and the generalized continuous wavelet transform associated with Laguerre functions. 9. Generalized Radon transforms on generalized hypergroups. 10. Inversion of generalized Radon transforms using generalized wavelets.

A bibliography at the end of the book gives a good impression of the area. One should also mention numerous publications by I. A. Kipriyanov and his collaborators from the Voronezh Mathematical School (Russia) devoted to generalized translation operators in different settings and the relevant harmonic analysis.

The book will be useful for researchers working in the area of harmonic analysis and its applications.

The author is an expert in this area and an initiator of studying wavelet transforms on hypergroups. The book consists of ten chapters: I. Product formulas and generalized hypergroups. 2. Hypergroups. 3. Wavelets and the windowed spherical Fourier transform on Gelfand pairs. 4. Generalized wavelets and generalized continuous wavelet transforms on hypergroups. 5. Generalized wavelets and generalized continuous wavelet transforms on semisimple Lie groups and on Cartan motion groups. 6. Harmonic analysis, generalized wavelets and the generalized continuous wavelet transforms on Chébli-Trimèche hypergroups. 7. Harmonic analysis, generalized wavelets and the generalized continuous wavelet transform associated with the spherical mean operator. 8. Harmonic analysis, generalized wavelets and the generalized continuous wavelet transform associated with Laguerre functions. 9. Generalized Radon transforms on generalized hypergroups. 10. Inversion of generalized Radon transforms using generalized wavelets.

A bibliography at the end of the book gives a good impression of the area. One should also mention numerous publications by I. A. Kipriyanov and his collaborators from the Voronezh Mathematical School (Russia) devoted to generalized translation operators in different settings and the relevant harmonic analysis.

The book will be useful for researchers working in the area of harmonic analysis and its applications.

Reviewer: B.Rubin (Jerusalem)

### MSC:

42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |

42-02 | Research exposition (monographs, survey articles) pertaining to harmonic analysis on Euclidean spaces |

43A62 | Harmonic analysis on hypergroups |

44A12 | Radon transform |

43-02 | Research exposition (monographs, survey articles) pertaining to abstract harmonic analysis |