The tread of this book is formed by two fundamental principles of
Harmonic Analysis: the Plancherel Formula and the Poisson S- mation
Formula. We ?rst prove both for locally compact abelian groups. For
non-abelian groups we 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 S- berg Trace Formula, which we
prove for arbitrary groups admitting uniform lattices. As examples for
the application of the Trace F- mula we treat the Heisenberg group and
the group SL (R). In the 2 2 former case the trace formula yields a
decomposition of the L -space of the Heisenberg group modulo a lattice.
In the case SL (R), the 2 trace formula is used to derive results like
the Weil asymptotic law for hyperbolic surfaces and to provide the
analytic continuation of the Selberg zeta function. We ?nally include a
chapter on the app- cations of abstract Harmonic Analysis on the theory
of wavelets. The present book is a text book for a graduate course on
abstract harmonic analysis and its applications. The book can be used as
a follow up of the First Course in Harmonic Analysis, [9], or indep-
dently, if the students have required a modest knowledge of Fourier
Analysis already. In this book, among other things, proofs are given of
Pontryagin Duality and the Plancherel Theorem for LCA-groups, which were
mentioned but not proved in [9].
