{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:I62PQHMK5KJQ73ZOVP2PSWG2UO","short_pith_number":"pith:I62PQHMK","canonical_record":{"source":{"id":"1204.4501","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","primary_cat":"math.NA","submitted_at":"2012-04-19T23:46:42Z","cross_cats_sorted":[],"title_canon_sha256":"ae91c1158316084c61952f44569e4ad586927bbd14874301f74533127c04654a","abstract_canon_sha256":"7d172b64faed7d211d6fdf2dedce29778edec55df1a8adee1ee54825b6978546"},"schema_version":"1.0"},"canonical_sha256":"47b4f81d8aea930fef2eabf4f958daa3b34ccdae1bfefc75ccf96f153bed661b","source":{"kind":"arxiv","id":"1204.4501","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1204.4501","created_at":"2026-05-18T03:44:13Z"},{"alias_kind":"arxiv_version","alias_value":"1204.4501v2","created_at":"2026-05-18T03:44:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1204.4501","created_at":"2026-05-18T03:44:13Z"},{"alias_kind":"pith_short_12","alias_value":"I62PQHMK5KJQ","created_at":"2026-05-18T12:27:09Z"},{"alias_kind":"pith_short_16","alias_value":"I62PQHMK5KJQ73ZO","created_at":"2026-05-18T12:27:09Z"},{"alias_kind":"pith_short_8","alias_value":"I62PQHMK","created_at":"2026-05-18T12:27:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:I62PQHMK5KJQ73ZOVP2PSWG2UO","target":"record","payload":{"canonical_record":{"source":{"id":"1204.4501","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","primary_cat":"math.NA","submitted_at":"2012-04-19T23:46:42Z","cross_cats_sorted":[],"title_canon_sha256":"ae91c1158316084c61952f44569e4ad586927bbd14874301f74533127c04654a","abstract_canon_sha256":"7d172b64faed7d211d6fdf2dedce29778edec55df1a8adee1ee54825b6978546"},"schema_version":"1.0"},"canonical_sha256":"47b4f81d8aea930fef2eabf4f958daa3b34ccdae1bfefc75ccf96f153bed661b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:44:13.536616Z","signature_b64":"T93F77gw0hpbK2rQ0+78Q94rLEcrtEcfcI2VlxZ7D5lx0rjtDgLnWY31wNRbkZwcbbptg1DEVIGMnRV471KBBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"47b4f81d8aea930fef2eabf4f958daa3b34ccdae1bfefc75ccf96f153bed661b","last_reissued_at":"2026-05-18T03:44:13.536196Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:44:13.536196Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1204.4501","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":"2026-05-18T03:44:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ElkZGOD/9Ka583kKnABdAT/XPxo6iBX0OUluutcZoRwt1apFawjqv1eD8+6phLtBTO2GPdrlR0ykm6xdXEoeDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T12:58:07.337275Z"},"content_sha256":"8ecbd3ccabe66f974bf3d0e6469ab50e383eaf8f32323ab570076be89defd4c5","schema_version":"1.0","event_id":"sha256:8ecbd3ccabe66f974bf3d0e6469ab50e383eaf8f32323ab570076be89defd4c5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:I62PQHMK5KJQ73ZOVP2PSWG2UO","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Discrete Fourier Analysis and Chebyshev Polynomials with $G_2$ Group","license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Huiyuan Li, Jiachang Sun, Yuan Xu","submitted_at":"2012-04-19T23:46:42Z","abstract_excerpt":"The discrete Fourier analysis on the $30^{\\degree}$-$60^{\\degree}$-$90^{\\degree}$ triangle is deduced from the corresponding results on the regular hexagon by considering functions invariant under the group $G_2$, which leads to the definition of four families generalized Chebyshev polynomials. The study of these polynomials leads to a Sturm-Liouville eigenvalue problem that contains two parameters, whose solutions are analogues of the Jacobi polynomials. Under a concept of $m$-degree and by introducing a new ordering among monomials, these polynomials are shown to share properties of the ordi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.4501","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":"2026-05-18T03:44:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5TTRgLcLQmH+En4o9izcB+Jv6asI2mel0lcenM/s4wAf67eV4kIlkAG6gNfpqONYDbAHAgJYN5Ghvp/vQjZRCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T12:58:07.337629Z"},"content_sha256":"fef3d71892be7bad72ad53049c6e723a2708459a17f8a6912cfa190aa89f1fac","schema_version":"1.0","event_id":"sha256:fef3d71892be7bad72ad53049c6e723a2708459a17f8a6912cfa190aa89f1fac"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/I62PQHMK5KJQ73ZOVP2PSWG2UO/bundle.json","state_url":"https://pith.science/pith/I62PQHMK5KJQ73ZOVP2PSWG2UO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/I62PQHMK5KJQ73ZOVP2PSWG2UO/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":"2026-05-28T12:58:07Z","links":{"resolver":"https://pith.science/pith/I62PQHMK5KJQ73ZOVP2PSWG2UO","bundle":"https://pith.science/pith/I62PQHMK5KJQ73ZOVP2PSWG2UO/bundle.json","state":"https://pith.science/pith/I62PQHMK5KJQ73ZOVP2PSWG2UO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/I62PQHMK5KJQ73ZOVP2PSWG2UO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:I62PQHMK5KJQ73ZOVP2PSWG2UO","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":"7d172b64faed7d211d6fdf2dedce29778edec55df1a8adee1ee54825b6978546","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","primary_cat":"math.NA","submitted_at":"2012-04-19T23:46:42Z","title_canon_sha256":"ae91c1158316084c61952f44569e4ad586927bbd14874301f74533127c04654a"},"schema_version":"1.0","source":{"id":"1204.4501","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1204.4501","created_at":"2026-05-18T03:44:13Z"},{"alias_kind":"arxiv_version","alias_value":"1204.4501v2","created_at":"2026-05-18T03:44:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1204.4501","created_at":"2026-05-18T03:44:13Z"},{"alias_kind":"pith_short_12","alias_value":"I62PQHMK5KJQ","created_at":"2026-05-18T12:27:09Z"},{"alias_kind":"pith_short_16","alias_value":"I62PQHMK5KJQ73ZO","created_at":"2026-05-18T12:27:09Z"},{"alias_kind":"pith_short_8","alias_value":"I62PQHMK","created_at":"2026-05-18T12:27:09Z"}],"graph_snapshots":[{"event_id":"sha256:fef3d71892be7bad72ad53049c6e723a2708459a17f8a6912cfa190aa89f1fac","target":"graph","created_at":"2026-05-18T03:44:13Z","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":"The discrete Fourier analysis on the $30^{\\degree}$-$60^{\\degree}$-$90^{\\degree}$ triangle is deduced from the corresponding results on the regular hexagon by considering functions invariant under the group $G_2$, which leads to the definition of four families generalized Chebyshev polynomials. The study of these polynomials leads to a Sturm-Liouville eigenvalue problem that contains two parameters, whose solutions are analogues of the Jacobi polynomials. Under a concept of $m$-degree and by introducing a new ordering among monomials, these polynomials are shown to share properties of the ordi","authors_text":"Huiyuan Li, Jiachang Sun, Yuan Xu","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","primary_cat":"math.NA","submitted_at":"2012-04-19T23:46:42Z","title":"Discrete Fourier Analysis and Chebyshev Polynomials with $G_2$ Group"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.4501","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":"sha256:8ecbd3ccabe66f974bf3d0e6469ab50e383eaf8f32323ab570076be89defd4c5","target":"record","created_at":"2026-05-18T03:44:13Z","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":"7d172b64faed7d211d6fdf2dedce29778edec55df1a8adee1ee54825b6978546","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","primary_cat":"math.NA","submitted_at":"2012-04-19T23:46:42Z","title_canon_sha256":"ae91c1158316084c61952f44569e4ad586927bbd14874301f74533127c04654a"},"schema_version":"1.0","source":{"id":"1204.4501","kind":"arxiv","version":2}},"canonical_sha256":"47b4f81d8aea930fef2eabf4f958daa3b34ccdae1bfefc75ccf96f153bed661b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"47b4f81d8aea930fef2eabf4f958daa3b34ccdae1bfefc75ccf96f153bed661b","first_computed_at":"2026-05-18T03:44:13.536196Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:44:13.536196Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"T93F77gw0hpbK2rQ0+78Q94rLEcrtEcfcI2VlxZ7D5lx0rjtDgLnWY31wNRbkZwcbbptg1DEVIGMnRV471KBBg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:44:13.536616Z","signed_message":"canonical_sha256_bytes"},"source_id":"1204.4501","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8ecbd3ccabe66f974bf3d0e6469ab50e383eaf8f32323ab570076be89defd4c5","sha256:fef3d71892be7bad72ad53049c6e723a2708459a17f8a6912cfa190aa89f1fac"],"state_sha256":"ce4295cf7e3ee738df5b37271d17b596b368b76f4216aef83e9116436c7cf289"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"piyt5ZMhXqOFasL2XaNxWyz29MBl/U140/mpj8LFf4gat/xXFce2DtHhO7YSO6S7sC/3Adu5bl2rBt6rDxTpAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T12:58:07.339490Z","bundle_sha256":"0a279ba9a98ea86a6a03987a4623dc003c90f7c992ca4052a9c3e6eddd2a8b23"}}