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Dimension walks and Schoenberg spectral measures

Authors: D. J. Daley and E. Porcu
Journal: Proc. Amer. Math. Soc. 142 (2014), 1813-1824
MSC (2010): Primary 62M30; Secondary 42B10
Published electronically: February 11, 2014
MathSciNet review: 3168486
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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(\textup {d} u)$, $ t=\Vert{\textbf {x}}\Vert$, for some uniquely identified probability measure $ G_d$ on $ \mathbb{R}_+$ and $ \Omega _d(t) = {\textup {E}} ({\textup {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=\Vert {\textbf {x}}\Vert$. 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$.

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D. J. Daley
Affiliation: Department of Mathematics and Statistics, The University of Melbourne, Victoria 3010, Australia

E. Porcu
Affiliation: Department of Statistics, Universidad Federico Santa Maria, Avenida España 1680, Valparaíso, 2390123, Chile

Received by editor(s): November 8, 2011
Received by editor(s) in revised form: April 1, 2012, and June 1, 2012
Published electronically: February 11, 2014
Communicated by: Edward C. Waymire
Article copyright: © Copyright 2014 American Mathematical Society

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