Positive definite radial functions on a domain

Tianshi Lu

Statistics & Probability Letters, 2025, vol. 226, issue C

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.

Keywords: Positive definite radial function on balls; Compact two-point homogeneous space (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1016/j.spl.2025.110520

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