Dimension walks and Schoenberg spectral measures
Daryl J. Daley, Emilio Porcu · Proceedings of the American Mathematical Society · 2014
Schoenberg (1938) identified the class of positive definite radial (or isotropic) functions φ : R d ↦ R \varphi :\mathbb {R}^d\mapsto \mathbb {R} , φ ( 0 ) = 1 \varphi ({\textbf {0}})=1 , as having a representation φ ( x ) = ∫ R + Ω d ( t u ) G d ( d u ) \varphi ({\textbf {x}}) = \int _{\mathbb {R}_+}\Omega _d(tu)\,G_d(\mathrm {d} u) , t = ‖ x ‖ t=\|{\textbf {x}}\| , for some uniquely identified probability measure G d G_d on R + \mathbb {R}_+ and Ω d ( t ) = E