On Gaussian kernels on Hilbert spaces and kernels on hyperbolic spaces

Abstract

This paper describes the concepts of Strictly Positive Definite, Universal, Integrally Strictly Positive Definite, C0-Universal for the Gaussian kernel on a Hilbert space. As a consequence we obtain a similar characterization for an important family of kernels studied and developed by Schoenberg and also on a family of spatial-time kernels popular in geostatistics, the Gneiting class, and its generalizations. Either by using similar techniques, or by a direct consequence of the Gaussian kernel on Hilbert spaces, we characterize the same concepts for a family of kernels defined on a real hyperbolic space.