{"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"}