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

Abstract

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 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.

Positive Definite Functions on Complex Spheres and their Walks through Dimensions

“…In this paper, inspired by the work of Beatson and zu Castell [4], we study positive definite functions on complex unit spheres Ω 2q of C q . In particular, we provide walks through dimensions over complex spheres.…”

“…In fact, the weakest possible condition to be used in Theorem 1.1(ii) follows from Guella and Menegatto [18] and reads as follows: 4) for every N ≥ 1, j = 0, 1, . .…”

Positive definite functions on the unit sphere and integrals of Jacobi polynomials

“…There has been renewed interest in positive or strictly positive definite functions due to application in spatial statistics and approximation theory; see, for example, the recent survey [13], as well as [4,5,6,14,19]. Among known examples, the function g t,δ has the simplest structure and, for a fixed x 0 ∈ S d−1 , the function x → g t,δ (d(x, x 0 )) has the compact support on S d−1 and its support is precisely the spherical cap c(x, θ) := {x ∈ S d−1 : d(x, x k ) ≤ t}.…”

A Note on the Derivatives of Isotropic Positive Definite Functions on the Hilbert Sphere

“…In the last five years there has been a tremendous amount of publications stating new results on positive definite functions on spheres, see for example [1, 4-6, 9, 13, 17, 19, 20]. Isotropic positive definite functions are used in approximation theory, where they are often referred to as spherical radial basis functions [2,3,12,22] and are for example applied in geostatistics and physiology [8,14]. They are also of importance in statistics where they occur as correlation functions of homogeneous random fields on spheres [15].…”