Compactly-Supported Isotropic Covariances on Spheres Obtained from Matrix-Valued Covariances in Euclidean Spaces

Let p, d be positive integers, with d odd. Let ϕ: [0 , + ∞) → R^p×p 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 ϕ is compactly supported over [0 , π] , then we show that the restriction of ϕ to [0 , π] 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 ϕ. Further, when ϕ 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.