Analysis on Symmetric Cones (oxford Mathematical Monographs)

measure always exists on a commutative hypergroup K, and there always exists a 'Plancherel measure' on the dual space K of equivalence classes of irreducible representations of K, with respect to which Plancherel's formula holds for Fourier transforms. In contrast with the case of a locally compact abelian group, however, the Plancherel measure is not necessarily a Haar measure on K, and K need not have a hypergroup structure. The third chapter presents the general construction and properties of hypergroups of polynomials, and of hypergroups in which the convolutions are connected with solutions of Sturm-Liouville boundary problems. Detailed information is given for numerous concrete examples of these two important classes of hypergroups, concerning their duals and their Haar and Plancherel measures. The main topics of Chapters 4 and 5 are positive and negative definite functions and continuous convolution semigroups of probability measures on hypergroups. Included are Bochner's theorem, the Levy continuity theorem, Levy-Khinchine representations, the shift compactness theorem, and work on the embedding problem for infinitely divisible distributions. Chapter 6 includes random walks and the potential theory of convolution semigroups on commutative hypergroups. In Chapter 7 it is shown that although there is no binary operation on K, it is possible to use the hypergroup structure to define a randomised sum of independent K-valued random variables. Strong laws of large numbers and central limit theory are then considered for randomised sums of independent random variables on some commutative hypergroups. This carefully written research monograph by two major contributors to the field gives a systematic and comprehensive account of harmonic analysis and probability theory on hypergroups. It will be indispensable as a text or for reference to anyone interested in this rapidly developing subject.