{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:BMZHFDTU6CXXHFG43NQPX32PKK","short_pith_number":"pith:BMZHFDTU","canonical_record":{"source":{"id":"1710.08280","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2017-10-23T14:01:24Z","cross_cats_sorted":[],"title_canon_sha256":"d42c5ec5c3e8bc4ef4a3b1345c0e281e026e2792cf69b69641d47d45432439bf","abstract_canon_sha256":"213d140bdca06cae4daf56e86c01005b0ad5a06ada2a53d38c714c333adf66e2"},"schema_version":"1.0"},"canonical_sha256":"0b32728e74f0af7394dcdb60fbef4f529b5f535069c1327796c3d092f9aa1a53","source":{"kind":"arxiv","id":"1710.08280","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1710.08280","created_at":"2026-05-18T00:32:16Z"},{"alias_kind":"arxiv_version","alias_value":"1710.08280v1","created_at":"2026-05-18T00:32:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1710.08280","created_at":"2026-05-18T00:32:16Z"},{"alias_kind":"pith_short_12","alias_value":"BMZHFDTU6CXX","created_at":"2026-05-18T12:31:08Z"},{"alias_kind":"pith_short_16","alias_value":"BMZHFDTU6CXXHFG4","created_at":"2026-05-18T12:31:08Z"},{"alias_kind":"pith_short_8","alias_value":"BMZHFDTU","created_at":"2026-05-18T12:31:08Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:BMZHFDTU6CXXHFG43NQPX32PKK","target":"record","payload":{"canonical_record":{"source":{"id":"1710.08280","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2017-10-23T14:01:24Z","cross_cats_sorted":[],"title_canon_sha256":"d42c5ec5c3e8bc4ef4a3b1345c0e281e026e2792cf69b69641d47d45432439bf","abstract_canon_sha256":"213d140bdca06cae4daf56e86c01005b0ad5a06ada2a53d38c714c333adf66e2"},"schema_version":"1.0"},"canonical_sha256":"0b32728e74f0af7394dcdb60fbef4f529b5f535069c1327796c3d092f9aa1a53","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:32:16.816184Z","signature_b64":"nqrH3GEgw2vEBA3Qdln1JNQYHeA82NWuxdT4a4WMAtFjzVr3yy24Xj8q47HjbWYGGaV2BF4M8OFKo62k7/I/BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0b32728e74f0af7394dcdb60fbef4f529b5f535069c1327796c3d092f9aa1a53","last_reissued_at":"2026-05-18T00:32:16.815708Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:32:16.815708Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1710.08280","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:32:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Ob6hadZ7rsOcJ4boVaMnFN4J4oxbtnHZdJL9+wE/Gdxy279iieO7HeddIAer5lqtIKkqAIAR+xOyE7WNg/2WBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T10:48:49.969230Z"},"content_sha256":"0941e5a080a3d589976c8e97de8b98116a9445220a5ce03ab47fe0a7464dfbf2","schema_version":"1.0","event_id":"sha256:0941e5a080a3d589976c8e97de8b98116a9445220a5ce03ab47fe0a7464dfbf2"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:BMZHFDTU6CXXHFG43NQPX32PKK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Gabor frames in $\\ell^2(\\mathbf Z)$ and linear dependence","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Marzieh Hasannasab, Ole Christensen","submitted_at":"2017-10-23T14:01:24Z","abstract_excerpt":"We prove that an overcomplete Gabor frame in $ \\ell^2(\\mathbf Z)$ by a finitely supported sequence is always linearly dependent. This is a particular case of a general result about linear dependence versus independence for Gabor systems in $\\ell^2(\\mathbf Z)$ with modulation parameter $1/M$ and translation parameter $N$ for some $M,N\\in \\mathbf N,$ and generated by a finite sequence $g$ in $\\ell^2(\\mathbf Z)$ with $K$ nonzero entries."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.08280","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:32:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BDpTu9fEid15wIBFwmMBeqfxE+kHlUiLmawjfYCvLmg7eg8oSYrxsejJXuPzfcqmVeOD3jXU1qRwFmBCa5mjDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T10:48:49.969586Z"},"content_sha256":"63b17380a793dc566835ffb508076738e191b2a00af75339f6aa04fcb0e20fbe","schema_version":"1.0","event_id":"sha256:63b17380a793dc566835ffb508076738e191b2a00af75339f6aa04fcb0e20fbe"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BMZHFDTU6CXXHFG43NQPX32PKK/bundle.json","state_url":"https://pith.science/pith/BMZHFDTU6CXXHFG43NQPX32PKK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BMZHFDTU6CXXHFG43NQPX32PKK/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-04T10:48:49Z","links":{"resolver":"https://pith.science/pith/BMZHFDTU6CXXHFG43NQPX32PKK","bundle":"https://pith.science/pith/BMZHFDTU6CXXHFG43NQPX32PKK/bundle.json","state":"https://pith.science/pith/BMZHFDTU6CXXHFG43NQPX32PKK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BMZHFDTU6CXXHFG43NQPX32PKK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:BMZHFDTU6CXXHFG43NQPX32PKK","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":"213d140bdca06cae4daf56e86c01005b0ad5a06ada2a53d38c714c333adf66e2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2017-10-23T14:01:24Z","title_canon_sha256":"d42c5ec5c3e8bc4ef4a3b1345c0e281e026e2792cf69b69641d47d45432439bf"},"schema_version":"1.0","source":{"id":"1710.08280","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1710.08280","created_at":"2026-05-18T00:32:16Z"},{"alias_kind":"arxiv_version","alias_value":"1710.08280v1","created_at":"2026-05-18T00:32:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1710.08280","created_at":"2026-05-18T00:32:16Z"},{"alias_kind":"pith_short_12","alias_value":"BMZHFDTU6CXX","created_at":"2026-05-18T12:31:08Z"},{"alias_kind":"pith_short_16","alias_value":"BMZHFDTU6CXXHFG4","created_at":"2026-05-18T12:31:08Z"},{"alias_kind":"pith_short_8","alias_value":"BMZHFDTU","created_at":"2026-05-18T12:31:08Z"}],"graph_snapshots":[{"event_id":"sha256:63b17380a793dc566835ffb508076738e191b2a00af75339f6aa04fcb0e20fbe","target":"graph","created_at":"2026-05-18T00:32:16Z","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 prove that an overcomplete Gabor frame in $ \\ell^2(\\mathbf Z)$ by a finitely supported sequence is always linearly dependent. This is a particular case of a general result about linear dependence versus independence for Gabor systems in $\\ell^2(\\mathbf Z)$ with modulation parameter $1/M$ and translation parameter $N$ for some $M,N\\in \\mathbf N,$ and generated by a finite sequence $g$ in $\\ell^2(\\mathbf Z)$ with $K$ nonzero entries.","authors_text":"Marzieh Hasannasab, Ole Christensen","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2017-10-23T14:01:24Z","title":"Gabor frames in $\\ell^2(\\mathbf Z)$ and linear dependence"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.08280","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":"sha256:0941e5a080a3d589976c8e97de8b98116a9445220a5ce03ab47fe0a7464dfbf2","target":"record","created_at":"2026-05-18T00:32:16Z","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":"213d140bdca06cae4daf56e86c01005b0ad5a06ada2a53d38c714c333adf66e2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2017-10-23T14:01:24Z","title_canon_sha256":"d42c5ec5c3e8bc4ef4a3b1345c0e281e026e2792cf69b69641d47d45432439bf"},"schema_version":"1.0","source":{"id":"1710.08280","kind":"arxiv","version":1}},"canonical_sha256":"0b32728e74f0af7394dcdb60fbef4f529b5f535069c1327796c3d092f9aa1a53","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0b32728e74f0af7394dcdb60fbef4f529b5f535069c1327796c3d092f9aa1a53","first_computed_at":"2026-05-18T00:32:16.815708Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:32:16.815708Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"nqrH3GEgw2vEBA3Qdln1JNQYHeA82NWuxdT4a4WMAtFjzVr3yy24Xj8q47HjbWYGGaV2BF4M8OFKo62k7/I/BA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:32:16.816184Z","signed_message":"canonical_sha256_bytes"},"source_id":"1710.08280","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0941e5a080a3d589976c8e97de8b98116a9445220a5ce03ab47fe0a7464dfbf2","sha256:63b17380a793dc566835ffb508076738e191b2a00af75339f6aa04fcb0e20fbe"],"state_sha256":"2f9723307f289cfd5a3ea5d7a3acf66af010e839267fa0149d8790268f3b6a8e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HiaGNRPLsZDh73NO1Vit5QnIk91b3eTYw9XuhaVQG7rZ4TDG31n/7JmuG9l8lWTU92if5pERlnsGmMmmIz1bAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-04T10:48:49.971476Z","bundle_sha256":"3fdc011e55b07d9d87c2dfdc3ee5fbf17bcd47e6434cf598b51bb98ac4e4424b"}}