Journal article


DJ Daley, E Porcu

Proceedings of the American Mathematical Society | AMER MATHEMATICAL SOC | Published : 2014


Schoenberg (1938) identified the class of positive definite radial (or isotropic) functions φ{symbol}: ℝ → ℝ, φ{symbol}(0) = 1, as having a representation φ{symbol}(x) = ∫ Ωd(tu)G (du), t = ||x||, for some uniquely identified probability measure G on ℝ and Ω (t) = E(e ), where η is a vector uniformly distributed on the unit spherical shell S ⊂ ℝ and e is a fixed unit vector. Call such G a d-Schoenberg measure, and let Φ 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 into Φ for some di..

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University of Melbourne Researchers


Awarded by Spanish Ministry of Science and Education

Funding Acknowledgements

The first author's work was done in part during visits to University Jaume I at Castellon, supported by grant MTM2010-14961 from the Spanish Ministry of Science and Education through Professor Jorge Mateu, and to the University of Sassari, where most of the second author's work was done. Both authors were supported at the University of Sassari by a grant from Regione Sardegna, Rientro Cervelli.