Principles of Harmonic Analysis (Universitext) 2nd Edition by Anton Deitmar (PDF)

Description

This book offers a complete and streamlined treatment of the central principles of abelian harmonic analysis: Pontryagin duality, the Plancherel theorem and the Poisson summation formula, as well as their respective generalizations to non-abelian groups, including the Selberg trace formula. The principles are then applied to spectral analysis of Heisenberg manifolds and Riemann surfaces. This new edition contains a new chapter on p-adic and adelic groups, as well as a complementary section on direct and projective limits. Many of the supporting proofs have been revised and refined. The book is an excellent resource for graduate students who wish to learn and understand harmonic analysis and for researchers seeking to apply it.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This book is closest to Rudin’s “Fourier Analysis on Groups” and Folland’s “A Course in Abstract Harmonic Analysis”. A reader must be comfortable with point set topology and measure theory. The first chapter introduces topological groups, Haar measure, and integration. The second chapter presents the Gelfand theory of commutative Banach algebras. The third and fourth chapters are about locally compact abelian groups and the Fourier transform. The fifth chapter is about operators on Hilbert spaces, and includes Hilbert-Schmidt and trace class operators, which are useful classes of operators when working with integral operators. The sixth chapter is the representation theory of locally compact groups. The seventh chapter proves the Peter-Weyl theorem for compact groups. The eighth chapter introduces the notion of a direct integral and von Neumann algebras. The ninth chapter proves the Selberg trace formula. The tenth chapter covers the Heisenberg group. The eleventh chapter is about SL(2,R), and includes Weyl’s asymptotic formula. The twelfth chapter is about wavelets. Finally, the thirteenth chapter is about p-adic numbers and adeles.This is an ideal book for a student of modular forms to learn harmonic analysis and functional analysis. I am especially pleased with the presentation of integral operators and p-adic numbers in this book. There are some technicalities that come up when rigorously defining integral operators that are ignored in almost every analysis book I have seen and these details are worked out better here than any other book I know. Likewise, the chapter on p-adic numbers and adeles carefully defines all the objects, like the restricted product topology on the adeles. In fact, I think that this book is the most understandable definition of the adeles that I have found in any book except for the first chapter of Goldfeld and Hundley,

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