{"paper":{"title":"From Schoenberg coefficients to Schoenberg functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Christian Berg, Emilio Porcu","submitted_at":"2015-05-21T11:53:57Z","abstract_excerpt":"In his seminal paper, Schoenberg (1942) characterized the class P(S^d) of continuous functions f:[-1,1] \\to \\R such that f(\\cos \\theta) is positive definite over the product space S^d \\times S^d, with S^d being the unit sphere of \\R^{d+1} and \\theta being the great circle distance. In this paper, we consider the product space S^d \\times G, for G a locally compact group, and define the class P(S^d, G) of continuous functions f:[-1,1]\\times G \\to \\C such that f(\\cos \\theta, u^{-1}\\cdot v) is positive definite on S^d \\times S^d \\times G \\times G. This offers a natural extension of Schoenberg's Th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.05682","kind":"arxiv","version":2},"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"}