Abstract
We call Φd the class of continuous functions φ:[0,∞)→[0,∞) such that the radial function ψ(x):=φ(‖x‖),x∈Rd, is positive definite on Rd, for d a positive integer. We then introduce the generalized Askey class of functions φn,k,m(⋅):[0,∞)→[0,∞) and show for which values of n,k and m such a class belongs to the class Φd. We then show walks through dimensions for scale mixtures of members of the class Φd with respect to nonnegative bounded measures; in particular, we show that, for a given member of Φd, there exist some classes of measures whose associated scale mixture does not preserve the same isotropy index d and allows us to jump into another dimension d′ for the class Φd. These facts open surprising connections with the celebrated class of multiply monotone functions.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have