{"paper":{"title":"Generalizations of Schoenberg's theorem on positive definite kernels","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.CA","authors_text":"Juan C. Vera, Olga Kuryatnikova","submitted_at":"2019-04-03T02:28:36Z","abstract_excerpt":"The seminal theorem of I.J. Schoenberg characterizes positive definite (p.d.) kernels on the unit sphere $S^{n-1}$ 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 $S^{n-1}$ and a compact set. Our second result characterizes invariant p.d. kernels on the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.02538","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}