NEGATIVE DEFINITE FUNCTIONS AND CONVOLUTION SEMIGROUPS
OF PROBABILITY MEASURES ON A COMMUTATIVE HYPERGROUP
Walter R. Bloom
Herbert Heyer
Abstract: Corresponding to the definitions of positive definite functions there are various
approaches to defining negative definite functions on hypergroups. These range from the
obvious “pointwise” definition to axiomatization via the Schoenberg duality. Researchers in
this area have used definitions best suited to their immediate purposes. In this paper we
present a comprehensive treatment of negative definite functions on commutative
hypergroups, leading to convolution semigroups of probability measures and their
Lévy-Khintchine representation within the framework of commutative hypergroups on
subsets of Euclidean space.
2000 AMS Mathematics Subject Classification: Primary: -; Secondary: -;
Key words and phrases: -