On the Positive Definiteness of Certain Functions

Abstract

For a coinmutative senugoup (S, +, *) with involution and a function f : S 4 [O, m), the set S ( f ) of those p 2 0 such that f* is a positive definite function on S is a closed subsemigroup of [O, 00) containing 0. For S = (Hi, +, G* = -G) it may happen that S(f) = { kd : k E No } for some d>O,anditmayhappenthatS(f)={O}u[d,m)forsomed>O. I f a > 2 a n d i f S = ( Z , + , n * = -n ) and f ( n ) = e-lnlo or S = (NO, +,no = n ) and f ( n ) = e n U , then S(f) n (0, c) = 0 and [d, 00) C S(f) for some d 2 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 a 2 3.4. In the last section we give sinipler proofs of previously known results concerning the positive definiteness of Ge-llzllo on normed spaces. 1991 Mathematics Subjeci Classification. Primary 42A82, 43A35. A) Given 5' and f what can be said about the semigroup S(f)? B) Given S which subsemigroups S of IR+ have the form S = S(f) with some f ?Our main concern will be with A) in some special cases. The additive semigroup No of nonnegative integers will be endowed with the identical involution n* = n while in the other cases S will be a group and we consider the group involution s+ = -s. As far as we know B) is open even for lN0, H (the additive group of integers) or IR (the additive group of real numbers).We start with the case S = J R and show that semigroups of the formoccur in B). Apart from the cases S = (0) and S = IR,+ we know of no other examples. Note that if S(f) = I R ' then f is c.alled infinitely divisible. Such functions have been thoroughly investigated and play an important role, among others, in probability theory (see e.g. [L]).