{"paper":{"title":"Relevant sampling in finitely generated shift-invariant spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Hartmut F\\\"uhr, Jun Xian","submitted_at":"2014-10-17T08:55:40Z","abstract_excerpt":"We consider random sampling in finitely generated shift-invariant spaces $V(\\Phi) \\subset {\\rm L}^2(\\mathbb{R}^n)$ generated by a vector $\\Phi = (\\varphi_1,\\ldots,\\varphi_r) \\in {\\rm L}^2(\\mathbb{R}^n)^r$. Following the approach introduced by Bass and Gr\\\"ochenig, we consider certain relatively compact subsets $V_{R,\\delta}(\\Phi)$ of such a space, defined in terms of a concentration inequality with respect to a cube with side lengths $R$. Under very mild assumptions on the generators, we show that for $R$ sufficiently large, taking $O(R^n log(R^{n^2/\\alpha'}))$ many random samples (taken indep"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.4666","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"}