Abstract

Let p, d be positive integers, with d odd. Let (/) : [0, +infinity) -> Rpxp be the isotropic part of a matrix-valued and isotropic covariance function (a positive semidefinite matrix-valued function) that is defined over the d-dimensional Euclidean space. If (/) is compactly supported over [0, pi], then we show that the restriction of (/) to [0, pi] is the isotropic part of a matrix-valued covariance function defined on a d-dimensional sphere, where isotropy in this case means that the covariance function depends on the geodesic distance. Our result does not need any assumption of continuity for the mapping (/). Further, when (/) is continuous, we provide an analytical expression of the d-Schoenberg sequence associated with the compactly-supported covariance on the sphere, which only requires knowledge of the Fourier transform of its isotropic part, and illustrate with the Gauss hypergeometric covariance model, which encompasses the well-known spherical, cubic, Askey and generalized Wendland covariances, and with a hole effect covariance model. Special cases of the results presented in this paper have been provided by other authors in the past decade.