{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:7XGZ7NQHWD2ACR2Y5WOM427OWS","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":"663f9a2dc3d02b8c0d4f47e7772f9d579a107caab7e230ba77deb31d488811e3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-10-21T08:28:10Z","title_canon_sha256":"c8cd6501f1dbbf6c2079c8e0da8f85139e69ebda12fda12a8230ae60d2d69063"},"schema_version":"1.0","source":{"id":"1610.06697","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1610.06697","created_at":"2026-05-17T23:50:28Z"},{"alias_kind":"arxiv_version","alias_value":"1610.06697v2","created_at":"2026-05-17T23:50:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.06697","created_at":"2026-05-17T23:50:28Z"},{"alias_kind":"pith_short_12","alias_value":"7XGZ7NQHWD2A","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_16","alias_value":"7XGZ7NQHWD2ACR2Y","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_8","alias_value":"7XGZ7NQH","created_at":"2026-05-18T12:30:04Z"}],"graph_snapshots":[{"event_id":"sha256:40b8aa10249c1d3ce238877d13b756a1489e57011ef5b4afcf658f039fc8da8c","target":"graph","created_at":"2026-05-17T23:50:28Z","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 use the concept of reproducing pairs to study Gabor systems at critical density. First, we present a generalization of the Balian-Low theorem to the reproducing pairs setting. Then, we prove our main result that there exists a reproducing partner for the Gabor system of integer time-frequency shifts of the Gaussian. In other words, the coefficients for this Gabor expansion of a square integrable function can be calculated using inner products with an unstructured family of vectors in $ L^2(\\R)$. This solves the possibly last open question for this system.","authors_text":"Michael Speckbacher, Peter Balazs","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-10-21T08:28:10Z","title":"Reproducing pairs and Gabor systems at critical density"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.06697","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":"sha256:6b43456914e6ad5ebb82629e9aed0b72a85dc0935d4ea3472facd4a971c046ab","target":"record","created_at":"2026-05-17T23:50:28Z","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":"663f9a2dc3d02b8c0d4f47e7772f9d579a107caab7e230ba77deb31d488811e3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-10-21T08:28:10Z","title_canon_sha256":"c8cd6501f1dbbf6c2079c8e0da8f85139e69ebda12fda12a8230ae60d2d69063"},"schema_version":"1.0","source":{"id":"1610.06697","kind":"arxiv","version":2}},"canonical_sha256":"fdcd9fb607b0f4014758ed9cce6beeb4912f8b07f2c30c64de50c3797c5e5316","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fdcd9fb607b0f4014758ed9cce6beeb4912f8b07f2c30c64de50c3797c5e5316","first_computed_at":"2026-05-17T23:50:28.954847Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:50:28.954847Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"vYVlo49a5r331Ozfc1uaSNCYHOitj762i3MazadtPwKjziTowefML/1KP4P+JSIw1OhCO9LRSGdDpARoGDR8Ag==","signature_status":"signed_v1","signed_at":"2026-05-17T23:50:28.955384Z","signed_message":"canonical_sha256_bytes"},"source_id":"1610.06697","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6b43456914e6ad5ebb82629e9aed0b72a85dc0935d4ea3472facd4a971c046ab","sha256:40b8aa10249c1d3ce238877d13b756a1489e57011ef5b4afcf658f039fc8da8c"],"state_sha256":"919b4025505983688b55e57684a31de3a6fcb6a8765455f2826cc11ee86c74d1"}