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

Audrey Anne Terras is currently Professor Emerita of Mathematics at the University of California at San Diego.

From the book reviews: "The present edition accommodates all the materials of the old one and maintains its style and spirit. ... It will acquaint that audience with this beautiful area and provide it with enough resources for further study. We hope this book will reinforce interest and excitement in the subject called harmonic analysis on symmetric spaces and, like its previous edition, it will be an important addition to individual collections and all the libraries, undergraduate, graduate and research." (Rudra P. Sarkar, Mathematical Reviews, November, 2014) "An updated version of an established work of scholarship that has both stood the test of time, and remains very relevant and well-worth reading. The intended audience is very broad, and includes scientists outside of pure mathematics ... as well as mathematicians who wish to enter the indicated field and desire a very well-drawn road-map. ... And doing the exercises is a solid pedagogical experience that will inaugurate the novice as well as the interested 'visitor' into active areas of research." (Michael Berg, MAA Reviews, November, 2013)

• Chapter 1 Flat Space. Fourier Analysis on R^m..- 1.1 Distributions or Generalized Functions.- 1.2 Fourier Integrals.- 1.3 Fourier Series and the Poisson Summation Formula.- 1.4 Mellin Transforms, Epstein and Dedekind Zeta Functions.- 1.5 Finite Symmetric Spaces, Wavelets, Quasicrystals, Weyl’s Criterion for Uniform Distribution.- Chapter 2 A Compact Symmetric Space--The Sphere.- 2.1 Fourier Analysis on the Sphere.- 2.2 O(3) and R^3. The Radon Transform.- Chapter 3 The Poincaré Upper Half-Plane.- 3.1 Hyperbolic Geometry.- 3.2 Harmonic Analysis on H.- 3.3 Fundamental Domains for Discrete Subgroups Γ of G = SL(2, R).- 3.4 Modular of Automorphic Forms--Classical.- 3.5 Automorphic Forms--Not So Classical--Maass Waveforms.- 3.6 Modular Forms and Dirichlet Series. Hecke Theory and Generalizations.- References.- Index.