Bernoulli

  • Bernoulli
  • Volume 19, Number 4 (2013), 1327-1349.

Strictly and non-strictly positive definite functions on spheres

Tilmann Gneiting

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Abstract

Isotropic positive definite functions on spheres play important roles in spatial statistics, where they occur as the correlation functions of homogeneous random fields and star-shaped random particles. In approximation theory, strictly positive definite functions serve as radial basis functions for interpolating scattered data on spherical domains. We review characterizations of positive definite functions on spheres in terms of Gegenbauer expansions and apply them to dimension walks, where monotonicity properties of the Gegenbauer coefficients guarantee positive definiteness in higher dimensions. Subject to a natural support condition, isotropic positive definite functions on the Euclidean space $\mathbb{R} ^{3}$, such as Askey’s and Wendland’s functions, allow for the direct substitution of the Euclidean distance by the great circle distance on a one-, two- or three-dimensional sphere, as opposed to the traditional approach, where the distances are transformed into each other. Completely monotone functions are positive definite on spheres of any dimension and provide rich parametric classes of such functions, including members of the powered exponential, Matérn, generalized Cauchy and Dagum families. The sine power family permits a continuous parameterization of the roughness of the sample paths of a Gaussian process. A collection of research problems provides challenges for future work in mathematical analysis, probability theory and spatial statistics.

Article information

Source
Bernoulli, Volume 19, Number 4 (2013), 1327-1349.

Dates
First available in Project Euclid: 27 August 2013

Permanent link to this document
https://projecteuclid.org/euclid.bj/1377612854

Digital Object Identifier
doi:10.3150/12-BEJSP06

Mathematical Reviews number (MathSciNet)
MR3102554

Zentralblatt MATH identifier
1283.62200

Keywords
completely monotone covariance localization fractal index interpolation of scattered data isotropic locally supported multiquadric Pólya criterion radial basis function Schoenberg coefficients

Citation

Gneiting, Tilmann. Strictly and non-strictly positive definite functions on spheres. Bernoulli 19 (2013), no. 4, 1327--1349. doi:10.3150/12-BEJSP06. https://projecteuclid.org/euclid.bj/1377612854


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Supplemental materials

  • Supplementary material: Supplement to “Strictly and non-strictly positive definite functions on spheres”. Appendix A states and proves further criteria of Pólya type, thereby complementing Section 4.2. Appendix B studies an example that involves oscillating trigonometric and Bessel functions, as hinted at in Section 4.5. Appendix C describes open problems that aim to stimulate future research in areas ranging from harmonic analysis to spatial statistics.