Strictly and non-strictly positive definite functions on spheres

Details Strictly and non-strictly positive definite functions on spheres Strictly and non-strictly positive definite functions on spheres

2011-11-30

arxiv.org/abs/1111.7077v4

Mathematics

Probability

Scientific paper

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 $\real^3$, 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, Mat\'ern and generalized Cauchy families. The sine power family permits a continuous parameterization of the roughness of the sample paths of Gaussian processes on spheres. The paper closes with a set of sixteen research problems that provide challenges for future work in mathematical analysis, probability theory and spatial statistics.

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