By Thabane L.

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**Example text**

2. l) is the support of ll· The unit point mass concentrated at s is indicated by Bs; C(H) is the space of continuous complex-valued functions on H, and Cc(H) consists of all f in C(H) with compact support. We will also use the notation t1 = M(~ol = {a=~ n=O anBn : ~ n=O lanl < oo}. If M(H) is a Banach algebra with multiplication * (called a convolution), then (H,*) is a hypergroup if the following axioms are satisfied: H1. If ll and v are probability measures, then so is ll * v. H2. l,v) ~ ll * v is continuous from M(H) x M(H) into M(H) where M(H) is given the weak topology with respect to Cc(H).

13 Lemma Let f E Then L2 (K) . positive definite function on f * the first is a bounded continuous K For commutative hypergroups we have two definiteness f- defined via further notions of positive an appropriate axiomatisation of Bochner's theorem. 14 called Definition Let K be a commutative strongly positive definite if We denote by for some v E f is M+(KA) b PB (K) the space of all (necessarily continuous and bounded) strongly positive definite functions 24 f = ~ hypergroup . 15 Definition Let K be a commutative hypergroup, and A locally bounded measurable function fK positive definite if We denote pc s l (K) by for all f dl' ~ 0 space the s positive definite functions on K .

Yet more general, but more complicated conditions are also given). Even when the product formulas fail to give rise to hyperRemark: groups, they may still give rise to interesting convolution algebras if positivity is preserved. For instance the so-called Gelfand Levitan spaces defined and studied by Gebuhrer [16] still require a positive convolution, but have no restrictions on the support. These spaces still admit a rich harmonic analysis. , see [15]. A similar situation can be studied in the discrete case also (see, for instance, ~6] and ~7]).