Representations of negative definite functions on polynomial hypergroups
For commutative hypergroups there exist a Bochner theorem for bounded positive definite functions as well as a Lévy-Khinchin formula for lower bounded, negative definite functions. In this paper the authors derive corresponding results for functions on polynomial hypergroups without any boundedness conditions. In this case, the representing measure and the Lévy measure \(\mu\), respectively, are positive measures on \(\mathbb{R}\) which is regarded as the space of all semicharacters. These measures are usually not unique. Moreover, algebraic conditions are given for \(\text{supp }\mu\subset \mathbb{R}_+\). Proofs are based on a transfer of the problem from polynomial hypergroups to the semigroup \((\mathbb{Z},+)\).
scientific article; zbMATH DE number 1773431