{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2008:C3SM4NPH7JJNS7JNLTIGYERHTO","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":"d59a85862b8e01147df36a2855972c2b8accb76e7cc7b6e18ec153004003da8c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2008-09-26T14:35:54Z","title_canon_sha256":"765a205ca3b0c3f324428e0a9c33b48ad2a001c4753ca18b6d2eda117cf4e7ae"},"schema_version":"1.0","source":{"id":"0809.4647","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0809.4647","created_at":"2026-05-18T03:53:32Z"},{"alias_kind":"arxiv_version","alias_value":"0809.4647v1","created_at":"2026-05-18T03:53:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0809.4647","created_at":"2026-05-18T03:53:32Z"},{"alias_kind":"pith_short_12","alias_value":"C3SM4NPH7JJN","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_16","alias_value":"C3SM4NPH7JJNS7JN","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_8","alias_value":"C3SM4NPH","created_at":"2026-05-18T12:25:57Z"}],"graph_snapshots":[{"event_id":"sha256:4f2be4b6ac90437b928ee869d80308675301df8f4dc26baa5e94dcf907145490","target":"graph","created_at":"2026-05-18T03:53:32Z","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":"Frames for Fr\\'echet spaces $X_F$ with respect to Fr\\'echet sequence spaces $\\Theta_F$ are studied and conditions, implying series expansions in $X_F$ and $X_F^*$, are determined. If $\\{g_i\\}$ is a $\\Theta_0$-frame for $X_0$, we construct a sequence $\\{X_s\\}_{s\\in {\\mathbb N}_0}$, $X_s\\subset X_{s-1}$, $s\\in {\\mathbb N}$, for given $\\Theta_F$, respectively a sequence $\\{\\Theta_s\\}_{s\\in {\\mathbb N}_0}$, $\\Theta_s\\subset \\Theta_{s-1}$, $s\\in {\\mathbb N}$, for given $X_F$, so that $\\{g_i\\}$ is a pre-$F$-frame (or $F$-frame) for $X_F$ with respect to $\\Theta_F$ under different assumptions given o","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":"2008-09-26T14:35:54Z","title":"Series expansions in Fr\\'echet spaces and their duals; construction of Fr\\'echet frames"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0809.4647","kind":"arxiv","version":1},"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:ffda21a183cbdfe9b99e821ac987308376b2f1bbf5e5d934110a270abbddfbb2","target":"record","created_at":"2026-05-18T03:53:32Z","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":"d59a85862b8e01147df36a2855972c2b8accb76e7cc7b6e18ec153004003da8c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2008-09-26T14:35:54Z","title_canon_sha256":"765a205ca3b0c3f324428e0a9c33b48ad2a001c4753ca18b6d2eda117cf4e7ae"},"schema_version":"1.0","source":{"id":"0809.4647","kind":"arxiv","version":1}},"canonical_sha256":"16e4ce35e7fa52d97d2d5cd06c12279b9ee4b74a60f7b77e3532cb753bc46df9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"16e4ce35e7fa52d97d2d5cd06c12279b9ee4b74a60f7b77e3532cb753bc46df9","first_computed_at":"2026-05-18T03:53:32.142074Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:53:32.142074Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"utOC5iqnt/98REFhr4JOkDB8srtdhs21qXK93ArTL4bCi3AO//vgR8BmVahxNzNXGHbZK+w6j151dQyxnF5uDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:53:32.142951Z","signed_message":"canonical_sha256_bytes"},"source_id":"0809.4647","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ffda21a183cbdfe9b99e821ac987308376b2f1bbf5e5d934110a270abbddfbb2","sha256:4f2be4b6ac90437b928ee869d80308675301df8f4dc26baa5e94dcf907145490"],"state_sha256":"e336433e6717ea5a8faa1288630a648b219887c158719d5940fdee786c46df73"}