{"paper":{"title":"Extension of positive definite functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.SP","authors_text":"Palle Jorgensen, Robert Niedzialomski","submitted_at":"2012-12-13T04:22:11Z","abstract_excerpt":"Let $\\Omega\\subset\\mathbb{R}^n$ be an open, connected subset of $\\mathbb{R}^n$, and let $F\\colon\\Omega-\\Omega\\to\\mathbb{C}$, where $\\Omega-\\Omega=\\{x-y\\colon x,y\\in\\Omega\\}$, be a continuous positive definite function. We give necessary and sufficient conditions for $F$ to have an extension to a continuous positive definite function defined on the entire Euclidean space $\\mathbb{R}^n$. The conditions are formulated in terms of strong commutativity of a system of certain unbounded selfadjoint operators defined on a Hilbert space associated to $F$. When a positive definite function $F$ is extend"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.3047","kind":"arxiv","version":3},"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"}