{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2007:JDBPOBJU45CX6QUKHSJVVJZNHF","short_pith_number":"pith:JDBPOBJU","canonical_record":{"source":{"id":"0712.3093","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.NA","submitted_at":"2007-12-19T04:00:14Z","cross_cats_sorted":["cs.NA","math.CA"],"title_canon_sha256":"4be7c87f73b0f9e04c0129d232e972c0ca5642e94c852d1cf21c15f5ce08a4ef","abstract_canon_sha256":"6d2b589909033b6e3fd9d7556f9f6cd01ebde06ad76571a0c376d60481113e98"},"schema_version":"1.0"},"canonical_sha256":"48c2f70534e7457f428a3c935aa72d3947c9b38705d64c2a363d2cc55d6121a8","source":{"kind":"arxiv","id":"0712.3093","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0712.3093","created_at":"2026-06-03T22:06:03Z"},{"alias_kind":"arxiv_version","alias_value":"0712.3093v1","created_at":"2026-06-03T22:06:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0712.3093","created_at":"2026-06-03T22:06:03Z"},{"alias_kind":"pith_short_12","alias_value":"JDBPOBJU45CX","created_at":"2026-06-03T22:06:03Z"},{"alias_kind":"pith_short_16","alias_value":"JDBPOBJU45CX6QUK","created_at":"2026-06-03T22:06:03Z"},{"alias_kind":"pith_short_8","alias_value":"JDBPOBJU","created_at":"2026-06-03T22:06:03Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2007:JDBPOBJU45CX6QUKHSJVVJZNHF","target":"record","payload":{"canonical_record":{"source":{"id":"0712.3093","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.NA","submitted_at":"2007-12-19T04:00:14Z","cross_cats_sorted":["cs.NA","math.CA"],"title_canon_sha256":"4be7c87f73b0f9e04c0129d232e972c0ca5642e94c852d1cf21c15f5ce08a4ef","abstract_canon_sha256":"6d2b589909033b6e3fd9d7556f9f6cd01ebde06ad76571a0c376d60481113e98"},"schema_version":"1.0"},"canonical_sha256":"48c2f70534e7457f428a3c935aa72d3947c9b38705d64c2a363d2cc55d6121a8","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-03T22:06:03.677798Z","signature_b64":"g04xFpDgCoscAnm1sco3pTofwxwXN/+E1Slpv+Nuzky9QerSXbeDKGGKbtv1lx0HrD8eowgSs4Z63Oy/CU2tAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"48c2f70534e7457f428a3c935aa72d3947c9b38705d64c2a363d2cc55d6121a8","last_reissued_at":"2026-06-03T22:06:03.677387Z","signature_status":"signed_v1","first_computed_at":"2026-06-03T22:06:03.677387Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0712.3093","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":"2026-06-03T22:06:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pOaMr0DCCC4RSaV2G4v1v6lcwH7VeiYOpOADOqkdAignInlZoPGnSOWSNZaDh8GDCGUIKnuJ6A0bIiKR0BsrAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T12:36:13.292816Z"},"content_sha256":"5c4fa679431e11f83c1e74c1903f2b3fac40f959cea027011864ec4a2331c3cc","schema_version":"1.0","event_id":"sha256:5c4fa679431e11f83c1e74c1903f2b3fac40f959cea027011864ec4a2331c3cc"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2007:JDBPOBJU45CX6QUKHSJVVJZNHF","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Discrete Fourier analysis, Cubature and Interpolation on a Hexagon and a Triangle","license":"","headline":"","cross_cats":["cs.NA","math.CA"],"primary_cat":"math.NA","authors_text":"Huiyuan Li, Jiachang Sun, Yuan Xu","submitted_at":"2007-12-19T04:00:14Z","abstract_excerpt":"Several problems of trigonometric approximation on a hexagon and a triangle are studied using the discrete Fourier transform and orthogonal polynomials of two variables. A discrete Fourier analysis on the regular hexagon is developed in detail, from which the analysis on the triangle is deduced. The results include cubature formulas and interpolation on these domains. In particular, a trigonometric Lagrange interpolation on a triangle is shown to satisfy an explicit compact formula, which is equivalent to the polynomial interpolation on a planer region bounded by Steiner's hypocycloid. The Leb"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0712.3093","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/0712.3093/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-06-03T22:06:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WYD1iwFYNvYt3DSVf2WVJHtkIlaecXGcalx3o4nGRnUMjwc7+TQskpeieHBehnhiK7EAcPBcuujJfEiQCY1EBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T12:36:13.293184Z"},"content_sha256":"38cdf0b8dceee17c9f599a1eb02642b1c19afb0084fac013079dd2fdbe098f2b","schema_version":"1.0","event_id":"sha256:38cdf0b8dceee17c9f599a1eb02642b1c19afb0084fac013079dd2fdbe098f2b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JDBPOBJU45CX6QUKHSJVVJZNHF/bundle.json","state_url":"https://pith.science/pith/JDBPOBJU45CX6QUKHSJVVJZNHF/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JDBPOBJU45CX6QUKHSJVVJZNHF/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":"2026-06-09T12:36:13Z","links":{"resolver":"https://pith.science/pith/JDBPOBJU45CX6QUKHSJVVJZNHF","bundle":"https://pith.science/pith/JDBPOBJU45CX6QUKHSJVVJZNHF/bundle.json","state":"https://pith.science/pith/JDBPOBJU45CX6QUKHSJVVJZNHF/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JDBPOBJU45CX6QUKHSJVVJZNHF/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2007:JDBPOBJU45CX6QUKHSJVVJZNHF","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":"6d2b589909033b6e3fd9d7556f9f6cd01ebde06ad76571a0c376d60481113e98","cross_cats_sorted":["cs.NA","math.CA"],"license":"","primary_cat":"math.NA","submitted_at":"2007-12-19T04:00:14Z","title_canon_sha256":"4be7c87f73b0f9e04c0129d232e972c0ca5642e94c852d1cf21c15f5ce08a4ef"},"schema_version":"1.0","source":{"id":"0712.3093","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0712.3093","created_at":"2026-06-03T22:06:03Z"},{"alias_kind":"arxiv_version","alias_value":"0712.3093v1","created_at":"2026-06-03T22:06:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0712.3093","created_at":"2026-06-03T22:06:03Z"},{"alias_kind":"pith_short_12","alias_value":"JDBPOBJU45CX","created_at":"2026-06-03T22:06:03Z"},{"alias_kind":"pith_short_16","alias_value":"JDBPOBJU45CX6QUK","created_at":"2026-06-03T22:06:03Z"},{"alias_kind":"pith_short_8","alias_value":"JDBPOBJU","created_at":"2026-06-03T22:06:03Z"}],"graph_snapshots":[{"event_id":"sha256:38cdf0b8dceee17c9f599a1eb02642b1c19afb0084fac013079dd2fdbe098f2b","target":"graph","created_at":"2026-06-03T22:06:03Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/0712.3093/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Several problems of trigonometric approximation on a hexagon and a triangle are studied using the discrete Fourier transform and orthogonal polynomials of two variables. A discrete Fourier analysis on the regular hexagon is developed in detail, from which the analysis on the triangle is deduced. The results include cubature formulas and interpolation on these domains. In particular, a trigonometric Lagrange interpolation on a triangle is shown to satisfy an explicit compact formula, which is equivalent to the polynomial interpolation on a planer region bounded by Steiner's hypocycloid. The Leb","authors_text":"Huiyuan Li, Jiachang Sun, Yuan Xu","cross_cats":["cs.NA","math.CA"],"headline":"","license":"","primary_cat":"math.NA","submitted_at":"2007-12-19T04:00:14Z","title":"Discrete Fourier analysis, Cubature and Interpolation on a Hexagon and a Triangle"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0712.3093","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":"sha256:5c4fa679431e11f83c1e74c1903f2b3fac40f959cea027011864ec4a2331c3cc","target":"record","created_at":"2026-06-03T22:06:03Z","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":"6d2b589909033b6e3fd9d7556f9f6cd01ebde06ad76571a0c376d60481113e98","cross_cats_sorted":["cs.NA","math.CA"],"license":"","primary_cat":"math.NA","submitted_at":"2007-12-19T04:00:14Z","title_canon_sha256":"4be7c87f73b0f9e04c0129d232e972c0ca5642e94c852d1cf21c15f5ce08a4ef"},"schema_version":"1.0","source":{"id":"0712.3093","kind":"arxiv","version":1}},"canonical_sha256":"48c2f70534e7457f428a3c935aa72d3947c9b38705d64c2a363d2cc55d6121a8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"48c2f70534e7457f428a3c935aa72d3947c9b38705d64c2a363d2cc55d6121a8","first_computed_at":"2026-06-03T22:06:03.677387Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-03T22:06:03.677387Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"g04xFpDgCoscAnm1sco3pTofwxwXN/+E1Slpv+Nuzky9QerSXbeDKGGKbtv1lx0HrD8eowgSs4Z63Oy/CU2tAQ==","signature_status":"signed_v1","signed_at":"2026-06-03T22:06:03.677798Z","signed_message":"canonical_sha256_bytes"},"source_id":"0712.3093","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5c4fa679431e11f83c1e74c1903f2b3fac40f959cea027011864ec4a2331c3cc","sha256:38cdf0b8dceee17c9f599a1eb02642b1c19afb0084fac013079dd2fdbe098f2b"],"state_sha256":"01dd0e1bac7a6679078a9fba78fcd6815a14eef4e9ca1ff18da341e31ef5f1df"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/OjCRv8JIXNCg0JFoQGoJJ64irAz11gjjApDS5zgAiOQYsbvR5J4qqYAmuyx1WKGPLpoR9fokDapZjiYhfEzAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T12:36:13.295168Z","bundle_sha256":"7553299f3a7798d1797d2591ad5cf1b20bdce282fdc0696a7014be63905b160e"}}