One-Step Recurrences for Stationary Random Fields on the Sphere

Symmetry Integrability and Geometry-Methods and Applications

article

R.K. Beatson¹  W. zu Castell² 

[1] School of Mathematics and Statistics, University of Canterbury, Private Bag 4800;Scientific Computing Research Unit

关键词: positive definite zonal functions;    ultraspherical expansions;    fractional integration;    Gegenbauer polynomials;   

DOI  :  10.3842/SIGMA.2016.043

来源: National Academy of Science of Ukraine

【 摘 要 】

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 $\{C^\lambda_n\}$.

【 授权许可】

Unknown