Dimension walks and Schoenberg spectral measures

Abstract:

Schoenberg (1938) identified the class of positive definite radial (or isotropic) functions $\varphi :\mathbb {R}^d\mapsto \mathbb {R}$, $\varphi ({\textbf {0}})=1$, as having a representation $\varphi ({\textbf {x}}) = \int _{\mathbb {R}_+}\Omega _d(tu) G_d(\mathrm {d} u)$, $t=\|{\textbf {x}}\|$, for some uniquely identified probability measure $G_d$ on $\mathbb {R}_+$ and $\Omega _d(t) = {\mathrm {E}} ({\mathrm {e}} ^{it\langle {\textbf {e}} _1, \mathbf {\scriptstyle \eta } \rangle })$, where $\mathbf {\eta }$ is a vector uniformly distributed on the unit spherical shell $\mathbb {S} ^{d-1} \subset \mathbb {R}^d$ and ${\textbf {e}}_1$ is a fixed unit vector. Call such $G_d$ a d-Schoenberg measure, and let $\Phi _d$ denote the class of all functions $f: \mathbb {R}_+ \mapsto \mathbb {R}$ for which such a $d$-dimensional radial function $\varphi$ exists with $f(t) = \varphi ({\textbf {x}} )$ for $t=\| {\textbf {x}}\|$. Mathéron (1965) introduced operators ${\widetilde {I}}$ and ${\widetilde {D}}$, called Montée and Descente, that map suitable $f\in \Phi _d$ into $\Phi _{d’}$ 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 $\Phi _\infty$ of completely monotone functions, for which neither ${\widetilde {I}} f$ nor ${\widetilde {D}} f$ is well defined. Because $f\in \Phi _d$ implies $f\in \Phi _{d’}$ for $d’<d$, any $f\in \Phi _d$ has a $d’$-Schoenberg measure $G_{d’}$ for $1\le d’<d$ and finite $d\ge 2$. This paper identifies $G_{d’}$ in terms of $G_d$ via another ‘dimension walk’ relating the Fourier transforms $\Omega _{d’}$ and $\Omega _d$ that reflect projections on $\mathbb {R} ^{d’}$ within $\mathbb {R} ^d$. A study of the Euclid hat function shows the indecomposability of $\Omega _d$.