On the Positive Definiteness of Certain Functions

For a coinmutative senugoup (S, +, *) with involution and a function f : S → [0, ∞), the set S( f ) of those p ≥ 0 such that f P is a positive definite function on S is a closed subsemigroup of [0, ∞) containing 0. For S = (IR, +, x* = ‐x ) it may happen that S( f ) = { kd : k ∈ N 0 } for some d > 0, and it may happen that S( f ) = {0} ⊂ [ d , ∞) for some d > O. If α > 2 and if S = (ℤ, +, n* = ‐n) and f ( n ) = e −[n]α or S = (IN 0 , +, n* = n) and f ( n ) = e nα , then S ( f ) ∪ (0, c ) = ϕ and [ d , ∞) ⊂ S( f ) for some d ≥; c > 0. Although (with c maximal and d minimal) we have not been able to show c = d in all cases, this equality does hold if S = ℤ and α ≥ 3.4. In the last section we give sinipler proofs of previously known results concerning the positive definiteness of x → e −||x||α on normed spaces.