Strictly Positive Definite Functions Original Research Article

We give a complete characterization of the strictly positive definite functions on the real line. By Bochnerʹs theorem, this is equivalent to proving that if the separated sequence of real numbers {an} describes the points of discontinuity of a distribution function, there exists an almost periodic polynomial with the zeros {an}. We prove a useful necessary condition that every strictly normalized, positive definite functionfsatisfies |f(x)|<1 for allx≠0. It is a sufficient condition for strictly positive definiteness that if the carrier of a nonzero finite Borel measure on R is not a discrete set, then the Fourier–Stieltjes transformμofμis strictly positive definite.