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.

one has Let denote the right hand side of this identity. Write , where and . Likewise decompose w as . Then . The Peter-Weyl theorem implies that . To see this, we decompose into a direct sum of irreducibles, each equivalent to . It then suffices to assume that lie in the same summand, since otherwise we have . The result then follows from expressing in terms of an orthonormal basis of . The spaces and its orthocomplement are invariant under π, therefore . Finally, as is a direct sum of isotypes

(triangle inequality) Remark Every absolute value maps to 1, i.e., one has For a proof consider so and so that Lemma 13.1.1 If is an absolute value on the field K, then is a metric on K. Proof The map d is positive definite. It is symmetric, too, since Finally. it satisfies the triangle inequality, since for one has Examples 13.1.2 For the usual absolute value is an example. The discrete absolute value exists for every field and is given by The metric generated by this absolute value

pairwise disjoint sets, we define Then the set equals , as C is dense in f ( X ). As f is integrable, the set is of finite measure. Let . Then s n is a simple function. We show that the sequence ( s n ) converges to f point-wise. Let . If , then for every n . So suppose . Then for some . For every one has , so that for each there exists a unique ν 0 with , hence and , which implies as claimed. We also see that by construction. So we get point-wise and , so by dominated convergence, Corollary

subspace of the algebraic dual space consisting of all linear functionals on V . We say that separates points in V if for any two in V there exists with . In this book we generally deal with vector spaces over . Sometimes it is convenient, however, to use vector spaces over instead. Every complex vector spaces naturally is a vector space over the reals. We will now show that in this case every real linear functional is the real part of a complex linear functional. Lemma C.1.1 Let W be a

given can be written uniquely as for and . Then The uniqueness of w is clear. Finally, . It is a bit disturbing that the canonical isomorphism is antilinear. It is possible to give linear isomorphisms, but not a canonical one. For this choose an orthonormal basis ( e i ) of V and define by . It is easy to see that this indeed is an isomorphism. Definition. Let I be an index set and for each fix a Hilbert space V i . The algebraic direct sum has a natural inner product for . If for