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

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Abstract

Let p, d be positive integers, with d odd. Let \(\varvec{\phi }:[0,+\infty ) \rightarrow {\mathbb {R}}^{p \times p}\) 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 \(\varvec{\phi }\) is compactly supported over \([0,\pi ]\), then we show that the restriction of \(\varvec{\phi }\) 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 \(\varvec{\phi }\). Further, when \(\varvec{\phi }\) 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.

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Acknowledgements

This work was supported by the National Agency for Research and Development of Chile [grants ANID/FONDECYT/REGULAR/No. 1210050 and ANID PIA AFB220002, and postgraduate study scholarship ANID-Subdirección de Capital Humano/Doctorado Nacional/2022-21220317], and by Khalifa University of Science and Technology [grant FSU-2021-016]. The authors acknowledge two anonymous reviewers for their insightful comments and suggestions.

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Authors and Affiliations

  1. Department of Mining Engineering, University of Chile, Avenida Beauchef 850, 8370448, Santiago, Chile

    Xavier Emery, Nadia Mery & Farzaneh Khorram

  2. Advanced Mining Technology Center, University of Chile, Avenida Beauchef 850, 8370448, Santiago, Chile

    Xavier Emery & Nadia Mery

  3. Facultad de Ingeniería, Universidad Andres Bello, República 239, 8370146, Santiago, Chile

    Farzaneh Khorram

  4. Department of Mathematics, Khalifa University, Abu Dhabi, United Arab Emirates

    Emilio Porcu

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  1. Xavier Emery
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  2. Nadia Mery
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Correspondence to Emilio Porcu.

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Communicated by Edward. B. Saff.

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Emery, X., Mery, N., Khorram, F. et al. Compactly-Supported Isotropic Covariances on Spheres Obtained from Matrix-Valued Covariances in Euclidean Spaces. Constr Approx 58, 181–198 (2023). https://doi.org/10.1007/s00365-022-09603-3

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