Description:
for Volume I -- I Flat Space. Fourier Analysis on ?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 -- II A Compact Symmetric Space—The Sphere -- 2.1. Spherical Harmonics -- 2.2. 0(3) and ?3. The Radon Transform -- III The Poincaré Upper Half-Plane -- 3.1. Hyperbolic Geometry -- 3.2. Harmonic Analysis on H -- 3.3. Fundamental Domains for Discrete Subgroups ? of G = SL(2,?) -- 3.4. Automorphic Forms—Classical -- 3.5. Automorphic Forms—Not So Classical—Maass Waveforms -- 3.6. Automorphic Forms and Dirichlet Series. Hecke Theory and Generalizations -- 3.7. Harmonic Analysis on the Fundamental Domain. The Roelcke-Selberg Spectral Resolution of the Laplacian, and the Selberg Trace Formula.
Since its beginnings with Fourier (and as far back as the Babylonian astron omers), harmonic analysis has been developed with the goal of unraveling the mysteries of the physical world of quasars, brain tumors, and so forth, as well as the mysteries of the nonphysical, but no less concrete, world of prime numbers, diophantine equations, and zeta functions. Quoting Courant and Hilbert, in the preface to the first German edition of Methods of Mathematical Physics: "Recent trends and fashions have, however, weakened the connection between mathematics and physics. " Such trends are still in evidence, harmful though they may be. My main motivation in writing these notes has been a desire to counteract this tendency towards specialization and describe appli cations of harmonic analysis in such diverse areas as number theory (which happens to be my specialty), statistics, medicine, geophysics, and quantum physics. I remember being quite surprised to learn that the subject is useful. My graduate eduation was that of the 1960s. The standard mathematics graduate course proceeded from Definition 1. 1. 1 to Corollary 14. 5. 59, with no room in between for applications, motivation, history, or references to related work. My aim has been to write a set of notes for a very different sort of course.