  We characterize completely the Gneiting class of space-time covariance
functions and give more relaxed conditions on the involved functions. We then
show necessary conditions for the construction of compactly supported functions
of the Gneiting type. These conditions are very general since they do not
depend on the Euclidean norm. Finally, we discuss a general class of positive
definite functions, used for multivariate Gaussian random fields. For this
class, we show necessary criteria for its generator to be compactly supported.
