Analysis IV. Integration and spectral theory, harmonic analysis, the garden of modular delights. Translated from the French by Urmie Ray (Q2339764)

Volume IV of ``Analysis'' is a continuation of Volumes I, II, III and consists of two chapters: XI -- Integration and Fourier transform and XII -- The garden of modular delights. In Chapter XI, following the N. Bourbaki treatment of the integration theory, the author treats: the upper integral of a positive function, \(L^{p}\) spaces, measurable sets and functions, Lebesgue-Fubini's way, the Lebesgue-Nikodym theorem and continues with spectral decomposition of a Hilbert space, the commutative Fourier transform, unitary representation of locally compact groups. In Chapter XII the author treats: infinite series and products in number theory, the series \(\sum 1/\cos \pi nz\) and \(\sum \exp \left( \pi in^{2}z\right) \), the Dirichlet series \(L\left( s;\chi \right) \), elliptic functions, \( SL_{2}\left( \mathbb{R}\right) \) as a locally compact group, modular functions: the classical theory, Fuchsian groups, Hecke theory, \( SL_{2}\left( \mathbb{R}\right) \) as a Lie group. As in the preceding volumes, the author includes many interesting historical comments of great value for those interested in the history of mathematics. The author includes sometimes examples and exercises which help the reader to understand the\ theory presented. As the preceding volumes the textbook is of highest mathematical level.