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Mathematical analysis IV. Integration and spectral theory, harmonic analysis, the garden of modular delights. (Analyse mathématique IV. Intégration et théorie spectrale, analyse harmonique, le jardin des délices modulaires.) (French) Zbl 1030.28001
Berlin: Springer (ISBN 3-540-43841-6/pbk). xii, 599 p. (2003).
This is the fourth, largest and last volume of the treatise on mathematical analysis of Roger Godement. The vivid style remains at the level of the three preceding ones (cf. the reviews in Zbl 0908.26001, Zbl 0908.26002, Zbl 0987.30001, respectively), even if it looks more “politically correct”. Integration theory is developed according to the Bourbaki’s approach, that the author justifies in a passionate way, which may not convince everybody. It is followed by spectral theory, Fourier transforms in \(L^1\) and \(L^2,\) and unitary representations of locally compact groups. The last part, which occupies half of the book, is a delightful and interesting presentation of analytic number theory, elliptic functions, modular functions, Fuchsian groups, Hecke’s theory and \(SL_2(R)\), under the unusual title “the garden of modular delights or the opium of mathematicians”. As in the other volumes, historical comments and personal remarks, as well as a tasteful blend of classical and modern results, give to the book his original flavor. The whole set of volumes brings back the nice tradition of the French treatises of analysis, which have blossomed during the nineteenth and twentieth centuries.

28-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to measure and integration
46-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functional analysis
43-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to abstract harmonic analysis
11Fxx Discontinuous groups and automorphic forms