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Positive definite radial functions on a domain

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  • Lu, Tianshi

Abstract

In this paper, we studied continuous radial functions that are positive definite on a domain D in the Euclidean space Rd or a compact two-point homogeneous space Md. We showed that for D⊂Rd that contains d-balls of arbitrary radius, a radial function that is PD on D is PD on Rd. On the other hand, for any closed proper subset D⊂Md, there exists a radial function that is PD on D but not PD on Md. We derived some sufficient conditions in terms of spectral coefficients for a continuous radial function that is PD on D to be PD on Md. As an example, we explicitly constructed radial functions that are PD on the unit ball embedded in the unit sphere Sd by a distance preserving map, but not PD on Sd.

Suggested Citation

  • Lu, Tianshi, 2025. "Positive definite radial functions on a domain," Statistics & Probability Letters, Elsevier, vol. 226(C).
    • Handle: RePEc:eee:stapro:v:226:y:2025:i:c:s0167715225001658
    DOI: 10.1016/j.spl.2025.110520
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    References listed on IDEAS

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    1. Tianshi Lu & Chunsheng Ma & Yimin Xiao, 2023. "Strong Local Nondeterminism and Exact Modulus of Continuity for Isotropic Gaussian Random Fields on Compact Two-Point Homogeneous Spaces," Journal of Theoretical Probability, Springer, vol. 36(4), pages 2403-2425, December.
    2. Chunsheng Ma & Anatoliy Malyarenko, 2020. "Time-Varying Isotropic Vector Random Fields on Compact Two-Point Homogeneous Spaces," Journal of Theoretical Probability, Springer, vol. 33(1), pages 319-339, March.
    3. Lu, Tianshi & Leonenko, Nikolai & Ma, Chunsheng, 2020. "Series representations of isotropic vector random fields on balls," Statistics & Probability Letters, Elsevier, vol. 156(C).
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