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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1410.4666","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2014-10-17T08:55:40Z","cross_cats_sorted":[],"title_canon_sha256":"23b15c3bcf3a059c664d465e4da79b371ca0b25fa535909c5e73c0550f19233a","abstract_canon_sha256":"d775cf25487d913659245462f8f8a95b06c325592dd62d9d31f239c8fb587149"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:39:55.048616Z","signature_b64":"y9Hct/UL5AZeiBqfsHt4lFkfsIlY0uzPJ5lAv+QOkoUPc7hblS7XNmzP/M44Tvxj8+5pRNrvJWYNVu9lj04hBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6417c36dc1ce5f0d4cf620f5b61302bb00cfe4e5ed7b21cf5d0c38e1e4d832a2","last_reissued_at":"2026-05-18T02:39:55.048180Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:39:55.048180Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $\\Phi = (\\varphi_1,\\ldots,\\varphi_r) \\in {\\rm L}^2(\\mathbb{R}^n)^r$. 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