## Abstract

Schoenberg (1938) identified the class of positive definite radial (or isotropic) functions φ{symbol}: ℝ^{d} → ℝ, φ{symbol}(0) = 1, as having a representation φ{symbol}(x) = ∫_{ℝ+} Ωd(tu)G_{d}(du), t = ||x||, for some uniquely identified probability measure G_{d} on ℝ_{+} and Ω_{d}(t) = E(e^{it(e1,η)}), where η is a vector uniformly distributed on the unit spherical shell S^{d-1} ⊂ ℝ^{d} and e_{1} is a fixed unit vector. Call such G_{d} a d-Schoenberg measure, and let Φ_{d} denote the class of all functions f: ℝ_{+} → ℝ for which such a d-dimensional radial function φ{symbol} exists with f(t) = φ{symbol}(x) for t = ||x||. Mathéron (1965) introduced operators Ĩ and D̃, called Montée and Descente, that map suitable f ∈ Φ_{d} into Φ_{d'} for some different dimension d': Wendland described such mappings as dimension walks. This paper characterizes Mathéron's operators in terms of Schoenberg measures and describes functions, even in the class Φ∞ of completely monotone functions, for which neither Ĩf nor D̃f is well defined. Because f ∈ Φ_{d} implies f ∈ Φ_{d'} for d' < d, any f ∈ Φ_{d} has a d' Schoenberg measure G_{d'} for 1 ≤ d' < d and finite d ≥ 2. This paper identifies G_{d'} in terms of G_{d} via another 'dimension walk' relating the Fourier transforms Ω_{d'} and Ω_{d} that reflect projections on ℝ^{d'} within ℝ^{d}. A study of the Euclid hat function shows the indecomposability of Ω_{d}.

Original language | British English |
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Pages (from-to) | 1813-1824 |

Number of pages | 12 |

Journal | Proceedings of the American Mathematical Society |

Volume | 142 |

Issue number | 5 |

DOIs | |

State | Published - 2014 |