Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 043, 19 pages      arXiv:1601.07743      https://doi.org/10.3842/SIGMA.2016.043
Contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications

One-Step Recurrences for Stationary Random Fields on the Sphere

R.K. Beatson a and W. zu Castell bc
a) School of Mathematics and Statistics, University of Canterbury, Private Bag 4800, Christchurch, New Zealand
b) Scientific Computing Research Unit, Helmholtz Zentrum München, Ingolstädter Landstraße 1, 85764 Neuherberg, Germany
c) Department of Mathematics, Technische Universität München, Germany

Received January 28, 2016, in final form April 15, 2016; Published online April 28, 2016

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 $\{C^\lambda_n\}$.

Key words: positive definite zonal functions; ultraspherical expansions; fractional integration; Gegenbauer polynomials.

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