This unique text is an introduction to harmonic analysis on the
simplest symmetric spaces, namely Euclidean space, the sphere, and
the Poincare upper half plane. This book is intended for beginning
graduate students in mathematics or researchers in physics or
engineering. Written with an informal style, the book places an
emphasis on motivation, concrete examples, history, and, above all,
applications in mathematics, statistics, physics, and
engineering.
Many corrections and updates have been incorporated in this new
edition. Updates include discussions of P. Sarnak and others' work
on quantum chaos, the work of T. Sunada, Marie-France Vigneras,
Carolyn Gordon, and others on Mark Kac's question "Can you hear the
shape of a drum?," A. Lubotzky, R. Phillips and P. Sarnak's
examples of Ramanujan graphs, and, finally, the author's
comparisons of continuous theory with the finite analogues.
Topics featured throughout the text include inversion formulas
for Fourier transforms, central limit theorems, Poisson's summation
formula and applications in crystallography and number theory,
applications of spherical harmonic analysis to the hydrogen atom,
the Radon transform, non-Euclidean geometry on the Poincare upper
half plane H or unit disc and applications to microwave
engineering, fundamental domains in H for discrete groups,
tessellations of H from such discrete group actions, automorphic
forms, and the Selberg trace formula and its applications in
spectral theory as well as number theory."
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