Generalizations of Schoenberg's theorem on positive definite kernels

Olga Kuryatnikova*, Juan C Vera

*Corresponding author for this work

Research output: Contribution to journalArticleAcademic

Abstract

The seminal theorem of I.J. Schoenberg characterizes positive definite (p.d.) kernels on the unit sphere Sn−1 invariant under the automorphisms of the sphere. We obtain two generalizations of this theorem for p.d. kernels on fiber bundles. Our first theorem characterizes invariant p.d. kernels on bundles whose fiber is a product of a compact set and the unit sphere. This result implies, in particular, a characterization of invariant under the automorphisms of the sphere p.d. kernels on a product of Sn−1 and a compact set. Our second result characterizes invariant p.d. kernels on the bundle whose fiber is Sn−1, base space is (Sn−1)r and map is the projection on the base space. This set of kernels is isomorphic to the set of invariant under the automorphisms of the sphere continuous functions F on (Sn−1)r+2 such that F(·,·,Z) is positive definite for every Z ∈ (Sn−1)r. When Z is fixed, this class reduces to the class of p.d. kernels invariant under the stabilizer of Z in the automorphism group of the sphere. For r = 1 these kernels have been used to obtain upper bounds for the spherical codes problem. Our extension for r > 1 can be used to construct new upper bounds on the size of spherical codes.
Original languageEnglish
Number of pages14
JournalarXiv preprint
Publication statusE-pub ahead of print - 2019
Externally publishedYes

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