Generalized Askey functions and their walks through dimensions

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.