Analysis on Symmetric Cones [Hardcover]

Analysis on Symmetric Conesis the first book to provide a systematic and clear introduction to the theory of symmetric cones, a subject of growing importance in number theory and multivariate analysis. Beginning with an elementary description of the Jordan algebra approach to the geometric and algebraic foundations of the theory, the book goes on to discuss harmonic analysis and special functions associated with symmetric cones, tying these results together with the study of holomorphic functions on bounded symmetric domains of tube type. Written by algebraic geometers, the book contains a detailed exposition of the spherical polynomials, multivariate hypergeometric functions, and invariant differential operators. The approach is based on Jordan algebras; all that is needed from the theory of these is developed in the first few chapters. The book will be read by students and theoreticians in pure mathematics, non-commutative harmonic analysis, Jordan algebras, and multivariate statistics.

1. Convex Cones
2. Jordan Algebras
3. Symmetric Cones and Euclidean Jordan Algebras
4. The Peirce Decomposition In a Jordan Algebra
5. Classification of Euclidean Jordan Algebras
6. Polar Decomposition and Gauss Decomposition
7. The Gamma Function of a Symmetric Cone
8. Complex Jordan Algebras
9. Tube Domains Over Convex cones
10. Symmetric Domains
11. Conical and Spherical Polynomials
12. Taylor and Laurent Series
13. Functions Spaces on Symmetric Domains
14. Invariant Differential Operators and Spherical Functions
15. Special Functions
16. Representations of Jordan Algebras and Euclidean Fourier Analysis
Bibliography

The detailed exposition, careful choice of organization and notation, and very helpful collection of exercises, mostly of medium difficulty, all attest to the effort put into this joint venture. --Bulletin of the London Mathematical Society

Whenever I have a problem or I wanl)