Positive definite functions, stationary covariance functions, and Bochner's theorem: Some results and a critical overview

In this article, the link among the classes of stationary covariance functions, positive definite functions, and the class of functions defined through Bochner's theorem has been properly analyzed: indeed, in the literature, the relationship among the above classes of functions has often generated some misunderstanding. Moreover, in order to provide an exhaustive outline on the above classes, a generalization of Bochner's theorem has been pointed out. For what concerns all the properties and results given in this article for positive definite and strictly positive definite functions, it has been underlined which of them are very general, that is, they are valid for any covariance function, hence they are independent from Bochner's theorem and which of them are strictly related to Bochner's representation, which provides a complete characterization for the special subclass of continuous covariance functions. On the other hand, several applications in time series, in spatial and, more generally, in spatiotemporal literature utilize covariance models which are characterized by a discontinuity at the origin, that is, a nugget effect, or are zero almost everywhere, hence these covariance functions cannot be represented through Bochner's theorem. Some useful results on the subset of the covariance functions, which are strictly positive definite and occur in all the interpolation problems, have also been given.