## Abstract

We call Φ_{d} 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 φ_{n,k,m}({dot operator}):[0, ∞)→[0, ∞) and show for which values of n, k and m such a class belongs to the class Φ_{d}. We then show walks through dimensions for scale mixtures of members of the class Φ_{d} with respect to nonnegative bounded measures; in particular, we show that, for a given member of Φ_{d}, 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 Φ_{d}. These facts open surprising connections with the celebrated class of multiply monotone functions.

Original language | British English |
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Pages (from-to) | 190-198 |

Number of pages | 9 |

Journal | Expositiones Mathematicae |

Volume | 32 |

Issue number | 2 |

DOIs | |

State | Published - 2014 |

## Keywords

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