{"paper":{"title":"Explicit formulas for $C^{1, 1}$ and $C^{1, \\omega}_{\\textrm{conv}}$ extensions of $1$-jets in Hilbert and superreflexive spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.FA","authors_text":"Carlos Mudarra, Daniel Azagra, Erwan Le Gruyer","submitted_at":"2017-06-07T15:52:19Z","abstract_excerpt":"Given $X$ a Hilbert space, $\\omega$ a modulus of continuity, $E$ an arbitrary subset of $X$, and functions $f:E\\to\\mathbb{R}$, $G:E\\to X$, we provide necessary and sufficient conditions for the jet $(f,G)$ to admit an extension $(F, \\nabla F)$ with $F:X\\to \\mathbb{R}$ convex and of class $C^{1, \\omega}(X)$, by means of a simple explicit formula. As a consequence of this result, if $\\omega$ is linear, we show that a variant of this formula provides explicit $C^{1,1}$ extensions of general (not necessarily convex) $1$-jets satisfying the usual Whitney extension condition, with best possible Lips"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.02235","kind":"arxiv","version":4},"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"}