Strictly positive definite functions

We give a complete characterization of the strictly positive definite Functions on the real line. By Bochner's theorem, this is equivalent to proving that if the separated sequence of real numbers {a(n)} describes the points of discontinuity of a distribution function, there exists an almost periodic polynomial with the zeros {a(n)}. We prove a useful necessary condition that every strictly normalized, positive definite function f satisfies \f(x)\ < 1 for all x not equal 0. It is a sufficient condition fur strictly positive definiteness that if the carrier of a nonzero finite Borel measure on R is not a discrete set, then the Fourier-Stieltjes transform <(mu)over cap> of mu is strictly positive definite. (C) 1996 Academic Press, Inc.