Generalized Askey functions and their walks through dimensions

Emilio Porcu, Viktor Zastavnyi

Research output: Contribution to journalArticlepeer-review

9 Scopus citations

Abstract

We call Φd the class of continuous functions φ:[0, ∞)→[0, ∞) such that the radial function ψ(x):=φ({norm of matrix}x{norm of matrix}),x∈Rd, is positive definite on Rd, for d a positive integer. We then introduce the generalized Askey class of functions φn,k,m({dot operator}):[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.

Original languageBritish English
Pages (from-to)190-198
Number of pages9
JournalExpositiones Mathematicae
Volume32
Issue number2
DOIs
StatePublished - 2014

Keywords

  • Askey functions
  • Multiply monotone functions
  • Positive definite radial functions
  • Primary
  • Secondary
  • Walks through dimensions

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