{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:PBSZ7Y7WAUY6TCWZDIQTKDZXEO","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"ce9df4dab1284d0895fa3d09e56c4bb483270af1a7c2067474038ecc37c9766e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-01-10T16:11:08Z","title_canon_sha256":"19667fc73f5957605a0db6c858b54dc2550bb98a4577ba900b6accace328574a"},"schema_version":"1.0","source":{"id":"1201.2096","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1201.2096","created_at":"2026-05-18T01:30:01Z"},{"alias_kind":"arxiv_version","alias_value":"1201.2096v2","created_at":"2026-05-18T01:30:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1201.2096","created_at":"2026-05-18T01:30:01Z"},{"alias_kind":"pith_short_12","alias_value":"PBSZ7Y7WAUY6","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_16","alias_value":"PBSZ7Y7WAUY6TCWZ","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_8","alias_value":"PBSZ7Y7W","created_at":"2026-05-18T12:27:18Z"}],"graph_snapshots":[{"event_id":"sha256:9e8320cf81f37c5bdb55cc5c3799c4cb1cc69b26aad6fbf932552232b1c87216","target":"graph","created_at":"2026-05-18T01:30:01Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We define an {\\it $(X_1,\\Theta, X_2)$-frame} with Banach spaces $X_2\\subseteq X_1$, $|\\cdot|_1 \\leq |\\cdot|_2$, and a $BK$-space $(\\Theta, \\snorm[\\cdot])$. Then by the use of decreasing sequences of Banach spaces ${X_s}_{s=0}^\\infty$ and of sequence spaces ${\\Theta_s}_{s=0}^\\infty$, we define a general Fr\\' echet frame on the Fr\\' echet space $X_F=\\bigcap_{s=0}^\\infty X_s$. We give frame expansions of elements of $X_F$ and its dual $X_F^*$, as well of some of the generating spaces of $X_F$ with convergence in appropriate norms. Moreover, we give necessary and sufficient conditions for a genera","authors_text":"Diana T. Stoeva, Stevan Pilipovi\\'c","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-01-10T16:11:08Z","title":"Fr\\'echet frames, general definition and expansions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.2096","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:b9ffc9a5a927794741fc7ea62be753714ad3c9dd64a58f58bc24153f4bc383b4","target":"record","created_at":"2026-05-18T01:30:01Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"ce9df4dab1284d0895fa3d09e56c4bb483270af1a7c2067474038ecc37c9766e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-01-10T16:11:08Z","title_canon_sha256":"19667fc73f5957605a0db6c858b54dc2550bb98a4577ba900b6accace328574a"},"schema_version":"1.0","source":{"id":"1201.2096","kind":"arxiv","version":2}},"canonical_sha256":"78659fe3f60531e98ad91a21350f3723adeeb1d52f2389a6bb4a7d4b1dc8d028","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"78659fe3f60531e98ad91a21350f3723adeeb1d52f2389a6bb4a7d4b1dc8d028","first_computed_at":"2026-05-18T01:30:01.426142Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:30:01.426142Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"WqS7KdWfs3Ifc9SAJc9FMiV3ZFHVITj2RElJ6KnedqKqTt5gO4SfOkNWrhTrVafH1tDWxHnMxZxOwb+TzSvTAw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:30:01.426584Z","signed_message":"canonical_sha256_bytes"},"source_id":"1201.2096","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b9ffc9a5a927794741fc7ea62be753714ad3c9dd64a58f58bc24153f4bc383b4","sha256:9e8320cf81f37c5bdb55cc5c3799c4cb1cc69b26aad6fbf932552232b1c87216"],"state_sha256":"6087d70432a925292529fa4b1615c7936efebcff199840dd32cfd4a20a829d3f"}