DIMENSION WALKS AND SCHOENBERG SPECTRAL MEASURES

Abstract

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..