Harmonic analysis on symmetric spaces and applications I-II /

Harmonic analysis, a subject of great maturity and vitality, has its roots in classical mathematics and has developed connections to many other branches of mathematical science, ranging from number theory and geometry to mathematical physics. If the subject has acquired the reputation of being difficult to approach for beginners, Terra's book should help to dispel that reputation. It is a motivated introduction that places ideas in their historical context, guides the reader to current research, and provides impressive applications of the theory to the real world (e.g., microwave engineering, NMR tomography, crystallography, quantum mechanics) and to other branches of mathematics (e.g., algebraic number fields, statistics, differential equations). The main topics treated are harmonic analysis on Euclidean space, the sphere, and the upper-half plane. This last topic makes up most of the present volume and introduces the theory of automorphic forms, including the Selberg trace formula. There are many examples and useful diagrams. The writing is suffused with Terra's personality and perspective; for all the information it imparts, it is never dry. Students at many levels may approach this book with profit; many sections can stand on their own. It takes both erudition and, dare one say, compassion to write mathematics this way; the book is a model of exposition and a testament to the spirit of the mathematical enterprise. Highly recommended for all libraries.-D.V. Feldman, Wesleyan University