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Principles of harmonic analysis. (English) Zbl 1158.43001

Universitext. New York, NY: Springer (ISBN 978-0-387-85468-7/pbk; 978-0-387-85469-4/ebook). xv, 333 p. (2009).
The book is intended as a text for a graduate course of abstract harmonic analysis and its applications. It consists of twelve chapters and their titles give a broad picture of the topics covered and their order of development:
1. Haar integration; 2. Banach algebras; 3. Duality for abelian groups; 4. The structure of LCA groups; 5. Operators on Hilbert spaces; 6. Representations; 7. Compact groups; 8. Direct integrals; 9. The Selberg trace formula; 10. The Heisenberg group; 11. \(\text{SL}_2(\mathbb R)\); 12. Wavelets.
Three appendices, on topology, measure and integration, and functional analysis, provide further necessary background. The book is based on two fundamental principles of harmonic analysis: the Plancherel formula and the Poisson summation formula. The authors first prove both for locally compact abelian groups. For non-abelian groups they discuss the Plancherel theorem in the general situation for type I groups. The generalization of the Poisson summation formula to non-abelian groups is the Selberg trace formula, which they prove for arbitrary groups admitting uniform lattices. As examples for the application of the trace formula the authors treat the Heisenberg group and the group \(\text{SL}_2(\mathbb R)\). In the former case the trace formula yields a decomposition of the \(L^2\)-space of the Heisenberg group modulo a lattice. In the case of \(\text{SL}_2(\mathbb R)\), the trace formula is used to derive results like the Weil asymptotic law for hyperbolic surfaces and to prove the analytic continuation of the Selberg zeta function. The book can be used as a follow-up to A. Deitmar’s previous book [A first course in harmonic analysis. Universitext. New York, NY: Springer (2005; Zbl 1063.43001)] or independently, if the reader has a modest knowledge of Fourier analysis.

MSC:

43-02 Research exposition (monographs, survey articles) pertaining to abstract harmonic analysis
22-02 Research exposition (monographs, survey articles) pertaining to topological groups
22B05 General properties and structure of LCA groups
22D25 \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
43A80 Analysis on other specific Lie groups

Citations:

Zbl 1063.43001
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