ON THE POSITIVE DEFINITENESS OF CERTAIN FUNCTIONS

For a commutative semigroup (S, +, ) with involution and a function f : S --> [0, infinity), the set S(f) of those p greater than or equal t o 0 such that f(p) is a positive definite function on S is a closed su bsemigroup of [0,infinity) containing 0. For S = (R, +, x() = -x) it may happen that S(f) = {kd : k epsilon N-0} for some d > 0, and it may happen that S(f) = {0} U [d,infinity) for some d > 0. If alpha > 2 an d if S = (Z, +, n() = -n) and f(n) = e(-\n\alpha) or S = (N-0,+,n(*) = n) and f(n) = e(n alpha), then S(f) boolean AND (0,c) = empty set an d [d, infinity) subset of S(f) for some d greater than or equal to c > 0. Although (with c maximal and d minimal) we have not been able to s how c = d in all cases, this equality does hold if S = Z and alpha gre ater than or equal to 3.4. In the last section we give simpler proofs of previously known results concerning the positive definiteness of x bar right arrow e (-parallel to x parallel to alpha) on normed spaces.