Dimension Walks on Generalized Spaces

Abstract

Let $d,k$ be positive integers. We call generalized spaces the cartesian product of the $d$-dimensional sphere, ${\mathbb{S}}^d$, with the $k$-dimensional Euclidean space, ${\mathbb{R}}^k$. We consider the class $\mathcal{P}({\mathbb{S}}^d\times {\mathbb{R}}^k)$ of continuous functions $\phi :[-1,1]\times [0,\infty )\to \mathbb{R}$ such that the mapping $C:{\left({\mathbb{S}}^d\times {\mathbb{R}}^k\right)}^2\to \mathbb{R}$, defined as $C((x,y),(x^{\prime },y^{\prime }))=\phi (\cos \theta (x,x^{\prime }),\Vert y-y^{\prime }\Vert )$, $(x,y),(x^{\prime },y^{\prime })\in {\mathbb{S}}^d\times {\mathbb{R}}^k$, is positive definite. We propose linear operators that allow for walks through dimension within generalized spaces while preserving positive definiteness.