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 t o proving that if the separated sequence of real numbers {a(n)} descri bes the points of discontinuity of a distribution function, there exis ts an almost periodic polynomial with the zeros {a(n)}. We prove a use ful necessary condition that every strictly normalized, positive defin ite function f satisfies \f(x)\ < 1 for all x not equal 0. It is a suf ficient condition fur strictly positive definiteness that if the carri er 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 posit ive definite. (C) 1996 Academic Press, Inc.