{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2007:O7U4REFSKBNCRW65B6EZWQ2CEM","short_pith_number":"pith:O7U4REFS","canonical_record":{"source":{"id":"0710.3652","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.AP","submitted_at":"2007-10-19T08:32:47Z","cross_cats_sorted":["math.NA"],"title_canon_sha256":"283a5d00b9a16e89dba3d2501b5cd1153248eefa7e2b2fb09367948713571a7b","abstract_canon_sha256":"0917a96bc4946e65721ad6279fbc69f5fb620ff367735c8c434c5f595c46cdac"},"schema_version":"1.0"},"canonical_sha256":"77e9c890b2505a28dbdd0f899b4342232605f7b6dc73f4324d29ff309512d2d4","source":{"kind":"arxiv","id":"0710.3652","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0710.3652","created_at":"2026-05-18T01:11:55Z"},{"alias_kind":"arxiv_version","alias_value":"0710.3652v1","created_at":"2026-05-18T01:11:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0710.3652","created_at":"2026-05-18T01:11:55Z"},{"alias_kind":"pith_short_12","alias_value":"O7U4REFSKBNC","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_16","alias_value":"O7U4REFSKBNCRW65","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_8","alias_value":"O7U4REFS","created_at":"2026-05-18T12:25:55Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2007:O7U4REFSKBNCRW65B6EZWQ2CEM","target":"record","payload":{"canonical_record":{"source":{"id":"0710.3652","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.AP","submitted_at":"2007-10-19T08:32:47Z","cross_cats_sorted":["math.NA"],"title_canon_sha256":"283a5d00b9a16e89dba3d2501b5cd1153248eefa7e2b2fb09367948713571a7b","abstract_canon_sha256":"0917a96bc4946e65721ad6279fbc69f5fb620ff367735c8c434c5f595c46cdac"},"schema_version":"1.0"},"canonical_sha256":"77e9c890b2505a28dbdd0f899b4342232605f7b6dc73f4324d29ff309512d2d4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:11:55.680038Z","signature_b64":"NzacNZ+RsYZwAuwwSgbjHVo+ogEcuUIQCcq5Q8IOJ9NCIOHSS74GF6cjaHXOMOmGrloyMLx+TDN/HyHJ2BB6DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"77e9c890b2505a28dbdd0f899b4342232605f7b6dc73f4324d29ff309512d2d4","last_reissued_at":"2026-05-18T01:11:55.679675Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:11:55.679675Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0710.3652","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":"2026-05-18T01:11:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xDyiN0mxmi4ocx1UjAOhgnZh8VenoBdlbMvE1zWqj4Otop6YkY94S9MLL1slR5dmXo4h3MFAFTmMqIPI1XvpAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T00:11:09.300710Z"},"content_sha256":"d252c19a1b218ee04838afea007ffc85bfefbb85248d7ad29cb67b496ecf128e","schema_version":"1.0","event_id":"sha256:d252c19a1b218ee04838afea007ffc85bfefbb85248d7ad29cb67b496ecf128e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2007:O7U4REFSKBNCRW65B6EZWQ2CEM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Time-Frequency Analysis of Fourier Integral Operators","license":"","headline":"","cross_cats":["math.NA"],"primary_cat":"math.AP","authors_text":"Elena Cordero, Fabio Nicola, Luigi Rodino","submitted_at":"2007-10-19T08:32:47Z","abstract_excerpt":"We use time-frequency methods for the study of Fourier Integral operators (FIOs). In this paper we shall show that Gabor frames provide very efficient representations for a large class of FIOs. Indeed, similarly to the case of shearlets and curvelets frames, the matrix representation of a Fourier Integral Operator with respect to a Gabor frame is well-organized. This is used as a powerful tool to study the boundedness of FIOs on modulation spaces. As special cases, we recapture boundedness results on modulation spaces for pseudo-differential operators with symbols in $M^{\\infty,1}$, for some u"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0710.3652","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":"2026-05-18T01:11:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bCeyFODcgqy/hcsGfOGYCq4xHA14W9FyduDR50uM6fdlr8kU8hWLLHe9T9IQ9CI+6gYy0tBZVvcf/mhl6kDGCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T00:11:09.301456Z"},"content_sha256":"67a1cf38eb9b5e136654701699713c19c4795838a422629f0840aea1150dfa50","schema_version":"1.0","event_id":"sha256:67a1cf38eb9b5e136654701699713c19c4795838a422629f0840aea1150dfa50"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/O7U4REFSKBNCRW65B6EZWQ2CEM/bundle.json","state_url":"https://pith.science/pith/O7U4REFSKBNCRW65B6EZWQ2CEM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/O7U4REFSKBNCRW65B6EZWQ2CEM/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":"2026-05-31T00:11:09Z","links":{"resolver":"https://pith.science/pith/O7U4REFSKBNCRW65B6EZWQ2CEM","bundle":"https://pith.science/pith/O7U4REFSKBNCRW65B6EZWQ2CEM/bundle.json","state":"https://pith.science/pith/O7U4REFSKBNCRW65B6EZWQ2CEM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/O7U4REFSKBNCRW65B6EZWQ2CEM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2007:O7U4REFSKBNCRW65B6EZWQ2CEM","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":"0917a96bc4946e65721ad6279fbc69f5fb620ff367735c8c434c5f595c46cdac","cross_cats_sorted":["math.NA"],"license":"","primary_cat":"math.AP","submitted_at":"2007-10-19T08:32:47Z","title_canon_sha256":"283a5d00b9a16e89dba3d2501b5cd1153248eefa7e2b2fb09367948713571a7b"},"schema_version":"1.0","source":{"id":"0710.3652","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0710.3652","created_at":"2026-05-18T01:11:55Z"},{"alias_kind":"arxiv_version","alias_value":"0710.3652v1","created_at":"2026-05-18T01:11:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0710.3652","created_at":"2026-05-18T01:11:55Z"},{"alias_kind":"pith_short_12","alias_value":"O7U4REFSKBNC","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_16","alias_value":"O7U4REFSKBNCRW65","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_8","alias_value":"O7U4REFS","created_at":"2026-05-18T12:25:55Z"}],"graph_snapshots":[{"event_id":"sha256:67a1cf38eb9b5e136654701699713c19c4795838a422629f0840aea1150dfa50","target":"graph","created_at":"2026-05-18T01:11:55Z","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 time-frequency methods for the study of Fourier Integral operators (FIOs). In this paper we shall show that Gabor frames provide very efficient representations for a large class of FIOs. Indeed, similarly to the case of shearlets and curvelets frames, the matrix representation of a Fourier Integral Operator with respect to a Gabor frame is well-organized. This is used as a powerful tool to study the boundedness of FIOs on modulation spaces. As special cases, we recapture boundedness results on modulation spaces for pseudo-differential operators with symbols in $M^{\\infty,1}$, for some u","authors_text":"Elena Cordero, Fabio Nicola, Luigi Rodino","cross_cats":["math.NA"],"headline":"","license":"","primary_cat":"math.AP","submitted_at":"2007-10-19T08:32:47Z","title":"Time-Frequency Analysis of Fourier Integral Operators"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0710.3652","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:d252c19a1b218ee04838afea007ffc85bfefbb85248d7ad29cb67b496ecf128e","target":"record","created_at":"2026-05-18T01:11:55Z","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":"0917a96bc4946e65721ad6279fbc69f5fb620ff367735c8c434c5f595c46cdac","cross_cats_sorted":["math.NA"],"license":"","primary_cat":"math.AP","submitted_at":"2007-10-19T08:32:47Z","title_canon_sha256":"283a5d00b9a16e89dba3d2501b5cd1153248eefa7e2b2fb09367948713571a7b"},"schema_version":"1.0","source":{"id":"0710.3652","kind":"arxiv","version":1}},"canonical_sha256":"77e9c890b2505a28dbdd0f899b4342232605f7b6dc73f4324d29ff309512d2d4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"77e9c890b2505a28dbdd0f899b4342232605f7b6dc73f4324d29ff309512d2d4","first_computed_at":"2026-05-18T01:11:55.679675Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:11:55.679675Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"NzacNZ+RsYZwAuwwSgbjHVo+ogEcuUIQCcq5Q8IOJ9NCIOHSS74GF6cjaHXOMOmGrloyMLx+TDN/HyHJ2BB6DQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:11:55.680038Z","signed_message":"canonical_sha256_bytes"},"source_id":"0710.3652","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d252c19a1b218ee04838afea007ffc85bfefbb85248d7ad29cb67b496ecf128e","sha256:67a1cf38eb9b5e136654701699713c19c4795838a422629f0840aea1150dfa50"],"state_sha256":"387476a77ea55d6c8747c211fc5fef79b549e115475a4859b6cca6c8f395dbea"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2/GaRi5s6NkveKBDwTBmQ1k+eDmgHSf43h6LXr2KcHphwHryGZBgA7e7Ay54h8rqLapJbnXI/zlmzzTfu7i4DA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T00:11:09.305299Z","bundle_sha256":"e90f8e4004551a4115b3e1b70fd386c24fa0d44192979c5881093bd35d5038ee"}}