Real and complex variable positive definite functions

In this paper we present an overview of the implications of our previously derived results for positive definite kernels on the general theory of positive definite functions. We begin by exploring the consequences of a set of differential inequalities on the global behaviour of a smooth positive definite function of one real variable. Then we propose a natural extension of this study to the complex variable case and derive consequences of positive definiteness for meromorphic functions. 1. Positive definite functions The purpose of this paper is to present a survey of some consequences of previously derived results on the theory of positive definite kernels for the field of positive definite functions. This a review paper; detailed proofs may be found where indicated. Received by the editors 2013-04-08. 2010Mathematics Subject Classification. Primary: 42A82. Secondary: 30A10, 30C40.