Dimension walks on hyperspheres

The seminal works by George Matheron provide the foundations of walks through dimensions for positive definite functions defined in Euclidean spaces. For a d-dimensional space and a class of positive definite functions therein, Matheron called montée and descente two operators that allow for obtaining new classes of positive definite functions in lower and higher dimensional spaces, respectively. The present work examines three different constructions to dimension walks for continuous positive definite functions on hyperspheres. First, we define montée and descente operators on the basis of the spectral representation of isotropic covariance functions on hyperspheres. The second approach provides walks through dimensions following Yadrenko’s construction of random fields on spheres. Under this approach, walks through unit dimensions are not permissible, while it is possible to walk under ±2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pm 2$$\end{document} dimensions from a covariance function that is valid on a d-dimensional sphere. The third construction relies on the integration of a given isotropic random field over latitudinal arcs. In each approach, we provide spectral representations of the montée and descente as well as illustrative examples.