{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:FN4ON4ACBEGQAJB4OB2VG7CK3Y","short_pith_number":"pith:FN4ON4AC","schema_version":"1.0","canonical_sha256":"2b78e6f002090d00243c7075537c4ade13deae0492263513768974a4f2e6e542","source":{"kind":"arxiv","id":"2605.04765","version":2},"attestation_state":"computed","paper":{"title":"A Generalized FC-Gram Approximation Framework with Analysis and Applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"A generalized FC-Gram framework adds flexibility to Gram polynomial blending and proves convergence rates of O(n to the minus min of r plus beta and d) for non-periodic functions.","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Akash Anand, Prakash Nainwal","submitted_at":"2026-05-06T11:12:36Z","abstract_excerpt":"The FC-Gram algorithm constructs high-order trigonometric approximations of nonperiodic functions by periodically extending them to a larger interval, with the quality of the blending continuation of Gram polynomials over the extension interval directly governing the approximation accuracy. We introduce GenFC, a generalized FC-Gram framework in which the continuation of each Gram polynomial is shaped by a cutoff function satisfying prescribed boundary flatness conditions. We establish a convergence theorem showing that for any such family the GenFC approximation error satisfies $O(n^{-\\min(r+\\"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2605.04765","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2026-05-06T11:12:36Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"a8c92f0c79f2f60b3999c0ba09f16e5a2c2c410325c610cff52f03df5af8dcfe","abstract_canon_sha256":"fee9ef7aaad0254ca9e14ce59d741cccff1224516b06a069185203fec748d31c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-10T01:10:02.754682Z","signature_b64":"O2rU7JWEJo9AM8oWunROOtDjgCtafW6Kgd2MYwBR3FyH90R6+s6kwyrO/okuQQk+/K0m7fpHKqpFdyGCpbT2Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2b78e6f002090d00243c7075537c4ade13deae0492263513768974a4f2e6e542","last_reissued_at":"2026-06-10T01:10:02.753849Z","signature_status":"signed_v1","first_computed_at":"2026-06-10T01:10:02.753849Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Generalized FC-Gram Approximation Framework with Analysis and Applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"A generalized FC-Gram framework adds flexibility to Gram polynomial blending and proves convergence rates of O(n to the minus min of r plus beta and d) for non-periodic functions.","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Akash Anand, Prakash Nainwal","submitted_at":"2026-05-06T11:12:36Z","abstract_excerpt":"The FC-Gram algorithm constructs high-order trigonometric approximations of nonperiodic functions by periodically extending them to a larger interval, with the quality of the blending continuation of Gram polynomials over the extension interval directly governing the approximation accuracy. We introduce GenFC, a generalized FC-Gram framework in which the continuation of each Gram polynomial is shaped by a cutoff function satisfying prescribed boundary flatness conditions. We establish a convergence theorem showing that for any such family the GenFC approximation error satisfies $O(n^{-\\min(r+\\"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We establish a convergence theorem showing that the trigonometric interpolant converges at the rate O(n^{-min(r+β,d)}) in the supremum norm on the original interval, where r is the smoothness of the target function, d the number of Gram polynomials, and β ∈ [0,1] a Fourier-decay parameter.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The blending continuation of Gram polynomials can be constructed with the stated flexibility while preserving the controlled boundary behavior and without introducing uncontrolled errors that would invalidate the convergence analysis.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"GenFC generalizes FC-Gram via flexible Gram polynomial blending, proving O(n^{-min(r+β,d)}) convergence and showing better accuracy than prior versions in numerical tests.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A generalized FC-Gram framework adds flexibility to Gram polynomial blending and proves convergence rates of O(n to the minus min of r plus beta and d) for non-periodic functions.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"f5547f1f13e0d6e67e2e93bf98d64e7c49837080ae0f06ebd4eed91c48c19dee"},"source":{"id":"2605.04765","kind":"arxiv","version":2},"verdict":{"id":"aaefa962-93b4-461d-80b4-4360bcb74ff8","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-08T15:47:49.606749Z","strongest_claim":"We establish a convergence theorem showing that the trigonometric interpolant converges at the rate O(n^{-min(r+β,d)}) in the supremum norm on the original interval, where r is the smoothness of the target function, d the number of Gram polynomials, and β ∈ [0,1] a Fourier-decay parameter.","one_line_summary":"GenFC generalizes FC-Gram via flexible Gram polynomial blending, proving O(n^{-min(r+β,d)}) convergence and showing better accuracy than prior versions in numerical tests.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The blending continuation of Gram polynomials can be constructed with the stated flexibility while preserving the controlled boundary behavior and without introducing uncontrolled errors that would invalidate the convergence analysis.","pith_extraction_headline":"A generalized FC-Gram framework adds flexibility to Gram polynomial blending and proves convergence rates of O(n to the minus min of r plus beta and d) for non-periodic functions."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.04765/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-20T11:34:43.324498Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T22:01:29.176049Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T14:11:40.207049Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"759834e52e7d2c7b5d2bbbf49217070d3f0b4c35bd3b2bf71e34137d868572a9"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2605.04765","created_at":"2026-06-10T01:10:02.753946+00:00"},{"alias_kind":"arxiv_version","alias_value":"2605.04765v2","created_at":"2026-06-10T01:10:02.753946+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.04765","created_at":"2026-06-10T01:10:02.753946+00:00"},{"alias_kind":"pith_short_12","alias_value":"FN4ON4ACBEGQ","created_at":"2026-06-10T01:10:02.753946+00:00"},{"alias_kind":"pith_short_16","alias_value":"FN4ON4ACBEGQAJB4","created_at":"2026-06-10T01:10:02.753946+00:00"},{"alias_kind":"pith_short_8","alias_value":"FN4ON4AC","created_at":"2026-06-10T01:10:02.753946+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/FN4ON4ACBEGQAJB4OB2VG7CK3Y","json":"https://pith.science/pith/FN4ON4ACBEGQAJB4OB2VG7CK3Y.json","graph_json":"https://pith.science/api/pith-number/FN4ON4ACBEGQAJB4OB2VG7CK3Y/graph.json","events_json":"https://pith.science/api/pith-number/FN4ON4ACBEGQAJB4OB2VG7CK3Y/events.json","paper":"https://pith.science/paper/FN4ON4AC"},"agent_actions":{"view_html":"https://pith.science/pith/FN4ON4ACBEGQAJB4OB2VG7CK3Y","download_json":"https://pith.science/pith/FN4ON4ACBEGQAJB4OB2VG7CK3Y.json","view_paper":"https://pith.science/paper/FN4ON4AC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2605.04765&json=true","fetch_graph":"https://pith.science/api/pith-number/FN4ON4ACBEGQAJB4OB2VG7CK3Y/graph.json","fetch_events":"https://pith.science/api/pith-number/FN4ON4ACBEGQAJB4OB2VG7CK3Y/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FN4ON4ACBEGQAJB4OB2VG7CK3Y/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FN4ON4ACBEGQAJB4OB2VG7CK3Y/action/storage_attestation","attest_author":"https://pith.science/pith/FN4ON4ACBEGQAJB4OB2VG7CK3Y/action/author_attestation","sign_citation":"https://pith.science/pith/FN4ON4ACBEGQAJB4OB2VG7CK3Y/action/citation_signature","submit_replication":"https://pith.science/pith/FN4ON4ACBEGQAJB4OB2VG7CK3Y/action/replication_record"}},"created_at":"2026-06-10T01:10:02.753946+00:00","updated_at":"2026-06-10T01:10:02.753946+00:00"}