arXiv subjects

R. K. Beatson

Publications and source records attributed to R. K. Beatson.

3 recordsLinked to original sources

arXiv2016

One-Step Recurrences for Stationary Random Fields on the Sphere

Recurrences for positive definite functions in terms of the space dimension have been used in several fields of applications. Such recurrences typically relate to properties of the system of special functions characterizing the geometry of the underlying space. In the case of the sphere ${\mathbb S}^{d-1} \subset {\mathbb R}^d$ the (strict) positive definiteness of the zonal function $f(\cos θ)$ is determined by the signs of the coefficients in the expansion of $f$ in terms of the Gegenbauer polynomials $\{C^λ_n\}$, with $λ=(d-2)/2$. Recent results show that classical differentiation and integration applied to $f$ have positive definiteness preserving properties in this context. However, in these results the space dimension changes in steps of two. This paper develops operators for zonal functions on the sphere which preserve (strict) positive definiteness while moving up and down in the ladder of dimensions by steps of one. These fractional operators are constructed to act appropriately on the Gegenbauer polynomials $\{C^λ_n\}$.

math.CA↗

arXiv2015

Dimension hopping and families of strictly positive definite zonal basis functions on spheres

Positive definite functions of compact support are widely used for radial basis function approximation as well as for estimation of spatial processes in geostatistics. Several constructions of such functions for ${\mathbb R}^d$ are based upon recurrence operators. These map functions of such type in a given space dimension onto similar ones in a space of lower or higher dimension. We provide analogs of these dimension hopping operators for positive definite, and strictly positive definite, zonal (radial) functions on the sphere. These operators are then used to provide new families of strictly positive definite functions with local support on the sphere.

math.CA↗

arXiv2011

A Pólya criterion for (strict) positive definiteness on the sphere

Positive definite functions are very important in both theory and applications of approximation theory, probability and statistics. In particular, identifying strictly positive definite kernels is of great interest as interpolation problems corresponding to these kernels are guaranteed to be poised. A Bochner type result of Schoenberg characterises continuous positive definite zonal functions, $f(\cos \cdot)$, on the sphere $\Sdmone$, as those with nonnegative Gegenbauer coefficients. More recent results characterise strictly positive definite functions on $\Sdmone$ by stronger conditions on the signs of the Gegenbauer coefficients. Unfortunately, given a function $f$, checking the signs of all the Gegenbauer coefficients can be an onerous, or impossible, task. Therefore, it is natural to seek simpler sufficient conditions which guarantee (strict) positive definiteness. We state a conjecture which leads to a Pólya type criterion for functions to be (strictly) positive definite on the sphere $\Sdmone$. In analogy to the case of the Euclidean space, the conjecture claims positivity of a certain integral involving Gegenbauer polynomials. We provide a proof of the conjecture for $d$ from 3 to 8.

math.CA↗