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A 'rational' analogue of the Ron-Shen Gramian is constructed, and prove that for any odd window function g the system G(g, {\\alpha}, {\\beta}) does not generate a frame if {\\alpha}{\\beta} = (n-1)/n. Special attention is paid to the first Hermite function h_1(t) = te^(-{\\pi}t^2)."},"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":"1108.2684","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2011-08-12T18:34:06Z","cross_cats_sorted":["math.IT"],"title_canon_sha256":"53c15ce8cd578a5a4a0b441bd71ca5937da0b881ce5bd172b4ebcd3517afaf0e","abstract_canon_sha256":"27273aa8e1e8eaa34d8bda0b9bbdcd6c03fca16c7526dece26ea1fc33e31f0bf"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:15:36.418760Z","signature_b64":"QVM2kjg8ty22MT+wK3hZehQfGJMwOaXwVJG6oaeoE5xJdmXOkoDKYkVCyGWrlndHivqMkbtwoc9vYFLvBtkiAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c409c4c74d6092205612107f545b9b9ac46c7292e270b3ca77361cf5d4c5c29c","last_reissued_at":"2026-05-18T04:15:36.418389Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:15:36.418389Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Gabor frames with rational density","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Preben Gr{\\aa}berg Nes, Yurii Lyubarskii","submitted_at":"2011-08-12T18:34:06Z","abstract_excerpt":"We consider the frame property of the Gabor system G(g, {\\alpha}, {\\beta}) = {e2{\\pi}i{\\beta}nt g(t - {\\alpha}m) : m, n \\in Z} for the case of rational oversampling, i.e. {\\alpha}, {\\beta} \\in Q. 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