{"paper":{"title":"Fourier representations of fractional B Splines via generalized Stirling type polynomials","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"Fractional B-splines admit a Fourier expansion in generalized Stirling-type numbers that represents them as infinite sums of Dirac delta derivatives in the distributional sense.","cross_cats":[],"primary_cat":"math.GM","authors_text":"Damla Gun, Peter Massopust, Yilmaz Simsek","submitted_at":"2026-05-14T07:43:08Z","abstract_excerpt":"In this paper, we investigate fractional B splines and their connections with Fourier analysis, and establish connections with generalized Stirling-type numbers and distribution theory. Employing a generating function approach inspired by recent results of Simsek [24], we derive a novel Fourier type expansion for fractional B splines that involves generalized Stirling type numbers. Our main contribution is the derivation of a Fourier-type expansion of fractional B splines in terms of generalized Stirling-type numbers. This representation allows us to express fractional B splines as infinite li"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Our main contribution is the derivation of a Fourier-type expansion of fractional B splines in terms of generalized Stirling-type numbers. This representation allows us to express fractional B splines as infinite linear combinations of derivatives of the Dirac delta in the distributional sense.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The generating function approach inspired by recent results of Simsek [24] extends directly to fractional B-splines to produce the claimed Fourier expansion and distributional representation (abstract, main contribution paragraph).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Derives Fourier representations of fractional B-splines via generalized Stirling-type polynomials, yielding distributional expressions with Dirac deltas and new fractional spline polynomials generated by the Mittag-Leffler function.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Fractional B-splines admit a Fourier expansion in generalized Stirling-type numbers that represents them as infinite sums of Dirac delta derivatives in the distributional sense.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"e3a2801c13f48100d3b46189408bb3c02aba835216b4608127cbc304b6cc6f68"},"source":{"id":"2605.15244","kind":"arxiv","version":1},"verdict":{"id":"56400b81-fbcd-404a-af0d-d4a233146e91","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T16:15:52.604929Z","strongest_claim":"Our main contribution is the derivation of a Fourier-type expansion of fractional B splines in terms of generalized Stirling-type numbers. This representation allows us to express fractional B splines as infinite linear combinations of derivatives of the Dirac delta in the distributional sense.","one_line_summary":"Derives Fourier representations of fractional B-splines via generalized Stirling-type polynomials, yielding distributional expressions with Dirac deltas and new fractional spline polynomials generated by the Mittag-Leffler function.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The generating function approach inspired by recent results of Simsek [24] extends directly to fractional B-splines to produce the claimed Fourier expansion and distributional representation (abstract, main contribution paragraph).","pith_extraction_headline":"Fractional B-splines admit a Fourier expansion in generalized Stirling-type numbers that represents them as infinite sums of Dirac delta derivatives in the distributional sense."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.15244/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T16:31:18.457390Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T16:27:11.061993Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T15:41:54.561887Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T13:33:22.821420Z","status":"skipped","version":"1.0.0","findings_count":0}],"snapshot_sha256":"0e609cb150f83f815a1af2452edba7f750e03ddd03718d845a9480704dc73dfd"},"references":{"count":28,"sample":[{"doi":"","year":2009,"title":"T. Agoh, K. Dilcher, Shortened recurrence relations for Bernoulli numbers, Discrete Math. 309 (2009) 887–898","work_id":"571baad6-f7fc-4611-83e8-b16b42a6060a","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1974,"title":"Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, Reidel, Dordrecht, 1974 (translated from French by J","work_id":"8cf32414-e0c4-43b6-8dd9-b36435e3aa06","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1937,"title":"Akhiezer, ¨Uber die beste Ann¨ aherung einer Klasse stetiger periodischer Funktionen, Dokl","work_id":"61efc51d-ca58-4c6b-9b6a-8259c8789ef2","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2000,"title":"T. Blu, M. Unser, Approximation error for quasi-interpolators and (multi-)wavelet expansions, Appl. Comput. Harmon. 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