Harmonic Analysis on Symmetric Spaces—Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane
2nd ed. 2013, XVII, 413 p. 83 illus., 32 illus. in color.
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Features many new examples and figures (such as density plots of modular forms as well as finite analogues)
New topics include: wavelets, quasicrystals, Ramanujan graphs, elliptic curves, modular knots, triangle and quaternion groups, comparisons of the continuous theory with its finite analogues, and recent progress on Mark Kac's question "Can you hear the shape of a drum?"
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.
Content Level »Graduate
Keywords »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