Positive Definite Functions on Products via Fourier Transforms: Old and New. (27th May 2022) - Menegatto, V. A.; Oliveira, C. P.
Abstract: A number of consistent models describing stationary positive definite functions on R m × R n stem from Bochner's celebrated theorem characterizing continuous and stationary positive definite functions on R m . If S m denotes the unit sphere in R m + 1, the same is true of positive definite functions on S m × R n which are radial with respect to the S m component and stationary with respect to the R n component. In this paper, we summarize results on these topics, mainly those that somehow characterize the positive definiteness of the function through Fourier transforms of the sections of the function itself. We present a new perspective on the existent results in the literature along with new characterizations and applications.