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 ∈ N₀ } for some d > 0, and it may happen that S(f) = {0} ? d, ∞) for some d > O. If α > 2 and if S = (?, +, n* = -n) and f(n) = e^?n]α or S = (IN₀, +, n* = n) and f(n) = e^nα, then S(f) ∪ (0, c) = ? 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 = ? 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.