Harmonic Analysis on Symmetric Spaces—Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane

  • Audrey Terras

Table of contents

  1. Front Matter
    Pages i-xvii
  2. Audrey Terras
    Pages 107-148
  3. Audrey Terras
    Pages 149-376
  4. Back Matter
    Pages 377-413

About this book


This unique text is an introduction to harmonic analysis on the simplest symmetric spaces, namely Euclidean space, the sphere, and the Poincaré 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, new topics, and updates have been incorporated in this new edition. These include discussions of the work of P. Sarnak and others making progress on various conjectures on modular forms, the work of T. Sunada, Marie-France Vignéras, Carolyn Gordon, and others on Mark Kac's question "Can you hear the shape of a drum?", Ramanujan graphs, wavelets, quasicrystals, modular knots, triangle and quaternion groups, computations of Maass waveforms, 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 Poincaré upper half plane H or unit disc and applications to microwave engineering, fundamental domains in H for discrete groups Γ, tessellations of H from such discrete group actions, automorphic forms, the Selberg trace formula and its applications in spectral theory as well as number theory.


Eisenstein series Hecke operators Helgason transform Poisson summation formula Schrödinger equation Selberg trace formula automorphic forms harmonic analysis modular forms modular knot quasicrystal zeta function

Authors and affiliations

  • Audrey Terras
    • 1
  1. 1.Department of MathematicsUniversity of California at San DiegoLa JollaUSA

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