{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:HLHMWO2QHSAEUDCDHXGJOJEWHZ","short_pith_number":"pith:HLHMWO2Q","canonical_record":{"source":{"id":"1207.5628","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-07-24T09:20:23Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"8ac33d8806c66c5a0bedb2cda82d7ab1a9520c7a4be4313dcb4948f4e3ba58e1","abstract_canon_sha256":"8e3673a62ed6cd72577e0ee4e4404d1ef8c33b13566553a9fb4de6c01f9ce5c8"},"schema_version":"1.0"},"canonical_sha256":"3acecb3b503c804a0c433dcc9724963e61fb889edd4fc15bf354a6a38ec6255f","source":{"kind":"arxiv","id":"1207.5628","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1207.5628","created_at":"2026-05-18T03:35:38Z"},{"alias_kind":"arxiv_version","alias_value":"1207.5628v2","created_at":"2026-05-18T03:35:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.5628","created_at":"2026-05-18T03:35:38Z"},{"alias_kind":"pith_short_12","alias_value":"HLHMWO2QHSAE","created_at":"2026-05-18T12:27:09Z"},{"alias_kind":"pith_short_16","alias_value":"HLHMWO2QHSAEUDCD","created_at":"2026-05-18T12:27:09Z"},{"alias_kind":"pith_short_8","alias_value":"HLHMWO2Q","created_at":"2026-05-18T12:27:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:HLHMWO2QHSAEUDCDHXGJOJEWHZ","target":"record","payload":{"canonical_record":{"source":{"id":"1207.5628","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-07-24T09:20:23Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"8ac33d8806c66c5a0bedb2cda82d7ab1a9520c7a4be4313dcb4948f4e3ba58e1","abstract_canon_sha256":"8e3673a62ed6cd72577e0ee4e4404d1ef8c33b13566553a9fb4de6c01f9ce5c8"},"schema_version":"1.0"},"canonical_sha256":"3acecb3b503c804a0c433dcc9724963e61fb889edd4fc15bf354a6a38ec6255f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:35:38.395690Z","signature_b64":"eRy62xLsnpYql9sC3CXW1AvVoDHxQiEPBnx9PSAXud17mEJZZhNQaQBbB/vuPnKqgNe1JMUuK6Q54r1aZYiODQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3acecb3b503c804a0c433dcc9724963e61fb889edd4fc15bf354a6a38ec6255f","last_reissued_at":"2026-05-18T03:35:38.394798Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:35:38.394798Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1207.5628","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:35:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4vyKTkbPU7jpJs0tiLTzvMsUaeL6sGB/Rqff9XQIVcSMGYH96qpUPTxJtyl9X7h98KsmkKnoP5P6zYRGMiE4AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T15:33:55.918629Z"},"content_sha256":"b37bf6722cfbee4062290fefe803fd9a189f13fc32604fb42b0622f0db51e054","schema_version":"1.0","event_id":"sha256:b37bf6722cfbee4062290fefe803fd9a189f13fc32604fb42b0622f0db51e054"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:HLHMWO2QHSAEUDCDHXGJOJEWHZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Gabor wave front set","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.FA","authors_text":"Luigi Rodino, Patrik Wahlberg","submitted_at":"2012-07-24T09:20:23Z","abstract_excerpt":"We define the Gabor wave front set $WF_G(u)$ of a tempered distribution $u$ in terms of rapid decay of its Gabor coefficients in a conic subset of the phase space. We show the inclusion $$WF_G(a^w(x,D) u) \\subseteq WF_G(u), u \\in \\mathscr S'(\\mathbb R^d), a \\in S_{0,0}^0,$$ where $S_{0,0}^0$ denotes the H\\\"ormander symbol class of order zero and parameter values zero. We compare our definition with other definitions in the literature, namely the classical and the global wave front sets of H\\\"ormander, and the $\\cS$-wave front set of Coriasco and Maniccia. In particular, we prove that the Gabor"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.5628","kind":"arxiv","version":2},"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":"2026-05-18T03:35:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gxGahVbmKTsRxNPqZ4ShfLtNr6wHf1Ye7ETZiF61QP+4l7dD54F/YnElIisE5IMvJcxu/0NKBBq+MOlhyuzaDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T15:33:55.919379Z"},"content_sha256":"da50cbf674a0f4d53f2f29c26d430bcf272cbcae18bd4a3ff89395959232ed7c","schema_version":"1.0","event_id":"sha256:da50cbf674a0f4d53f2f29c26d430bcf272cbcae18bd4a3ff89395959232ed7c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/HLHMWO2QHSAEUDCDHXGJOJEWHZ/bundle.json","state_url":"https://pith.science/pith/HLHMWO2QHSAEUDCDHXGJOJEWHZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/HLHMWO2QHSAEUDCDHXGJOJEWHZ/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":"2026-05-31T15:33:55Z","links":{"resolver":"https://pith.science/pith/HLHMWO2QHSAEUDCDHXGJOJEWHZ","bundle":"https://pith.science/pith/HLHMWO2QHSAEUDCDHXGJOJEWHZ/bundle.json","state":"https://pith.science/pith/HLHMWO2QHSAEUDCDHXGJOJEWHZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/HLHMWO2QHSAEUDCDHXGJOJEWHZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:HLHMWO2QHSAEUDCDHXGJOJEWHZ","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":"8e3673a62ed6cd72577e0ee4e4404d1ef8c33b13566553a9fb4de6c01f9ce5c8","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-07-24T09:20:23Z","title_canon_sha256":"8ac33d8806c66c5a0bedb2cda82d7ab1a9520c7a4be4313dcb4948f4e3ba58e1"},"schema_version":"1.0","source":{"id":"1207.5628","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1207.5628","created_at":"2026-05-18T03:35:38Z"},{"alias_kind":"arxiv_version","alias_value":"1207.5628v2","created_at":"2026-05-18T03:35:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.5628","created_at":"2026-05-18T03:35:38Z"},{"alias_kind":"pith_short_12","alias_value":"HLHMWO2QHSAE","created_at":"2026-05-18T12:27:09Z"},{"alias_kind":"pith_short_16","alias_value":"HLHMWO2QHSAEUDCD","created_at":"2026-05-18T12:27:09Z"},{"alias_kind":"pith_short_8","alias_value":"HLHMWO2Q","created_at":"2026-05-18T12:27:09Z"}],"graph_snapshots":[{"event_id":"sha256:da50cbf674a0f4d53f2f29c26d430bcf272cbcae18bd4a3ff89395959232ed7c","target":"graph","created_at":"2026-05-18T03:35:38Z","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 the Gabor wave front set $WF_G(u)$ of a tempered distribution $u$ in terms of rapid decay of its Gabor coefficients in a conic subset of the phase space. We show the inclusion $$WF_G(a^w(x,D) u) \\subseteq WF_G(u), u \\in \\mathscr S'(\\mathbb R^d), a \\in S_{0,0}^0,$$ where $S_{0,0}^0$ denotes the H\\\"ormander symbol class of order zero and parameter values zero. We compare our definition with other definitions in the literature, namely the classical and the global wave front sets of H\\\"ormander, and the $\\cS$-wave front set of Coriasco and Maniccia. In particular, we prove that the Gabor","authors_text":"Luigi Rodino, Patrik Wahlberg","cross_cats":["math.AP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-07-24T09:20:23Z","title":"The Gabor wave front set"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.5628","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:b37bf6722cfbee4062290fefe803fd9a189f13fc32604fb42b0622f0db51e054","target":"record","created_at":"2026-05-18T03:35:38Z","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":"8e3673a62ed6cd72577e0ee4e4404d1ef8c33b13566553a9fb4de6c01f9ce5c8","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-07-24T09:20:23Z","title_canon_sha256":"8ac33d8806c66c5a0bedb2cda82d7ab1a9520c7a4be4313dcb4948f4e3ba58e1"},"schema_version":"1.0","source":{"id":"1207.5628","kind":"arxiv","version":2}},"canonical_sha256":"3acecb3b503c804a0c433dcc9724963e61fb889edd4fc15bf354a6a38ec6255f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3acecb3b503c804a0c433dcc9724963e61fb889edd4fc15bf354a6a38ec6255f","first_computed_at":"2026-05-18T03:35:38.394798Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:35:38.394798Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"eRy62xLsnpYql9sC3CXW1AvVoDHxQiEPBnx9PSAXud17mEJZZhNQaQBbB/vuPnKqgNe1JMUuK6Q54r1aZYiODQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:35:38.395690Z","signed_message":"canonical_sha256_bytes"},"source_id":"1207.5628","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b37bf6722cfbee4062290fefe803fd9a189f13fc32604fb42b0622f0db51e054","sha256:da50cbf674a0f4d53f2f29c26d430bcf272cbcae18bd4a3ff89395959232ed7c"],"state_sha256":"0ac992d53e3d17e95454905d60bc35de55c62bb23dd8a5fa4922de118d71f2b8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YCpj7FuB5tZGdYXRU0heDODp34HT77vUrR1yLF8KSNJtChc84HqRTnE3BuN3Dy2i791btcrNVNRHFL9FPFP8BQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T15:33:55.924479Z","bundle_sha256":"aae285d36ba008195eed98ba42b0be949518f76e64d43937cdf7094b1c1d2d9d"}}