One-Step Recurrences for Stationary Random Fields on the Sphere

TL;DR

This paper introduces fractional operators that preserve positive definiteness of zonal functions on spheres while changing the dimension by one, extending previous results that only allowed dimension changes in steps of two.

Contribution

It develops new operators acting on Gegenbauer polynomial expansions that enable dimension adjustments by one while maintaining positive definiteness.

Findings

Operators preserve positive definiteness across dimensions

Operators act on Gegenbauer polynomial expansions

Dimension change steps are reduced from two to one

Abstract

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 $f(\cos \theta)$ is determined by the signs of the coefficients in the expansion of $f$ in terms of the Gegenbauer polynomials $\{C^\lambda_n\}$, with $\lambda=(d-2)/2$. 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 $\{C^\lambda_n\}$.