{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:31:40Z","timestamp":1760059900171,"version":"build-2065373602"},"reference-count":62,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2025,7,20]],"date-time":"2025-07-20T00:00:00Z","timestamp":1752969600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The paper presents an enhanced numerical framework for computing the one-dimensional fast Fractional Fourier Transform (FRFT) by integrating closed-form Composite Newton-Cotes quadrature rules. We show that a FRFT of a QN-length weighted sequence can be decomposed analytically into two mathematically commutative compositions: one involving the composition of a FRFT of an N-length sequence and a FRFT of a Q-length weighted sequence, and the other in reverse order. The composite FRFT approach is applied to the inversion of Fourier and Laplace transforms, with a focus on estimating probability densities for distributions with complex-valued characteristic functions. Numerical experiments on the Variance-Gamma (VG) and Generalized Tempered Stable (GTS) models show that the proposed scheme significantly improves accuracy over standard (non-weighted) fast FRFT and classical Newton-Cotes quadrature, while preserving computational efficiency. The findings suggest that the composite FRFT framework offers a robust and mathematically sound tool for transform-based numerical approximations, particularly in applications involving oscillatory integrals and complex-valued characteristic functions.<\/jats:p>","DOI":"10.3390\/axioms14070543","type":"journal-article","created":{"date-parts":[[2025,7,21]],"date-time":"2025-07-21T09:33:53Z","timestamp":1753090433000},"page":"543","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Enhanced Fast Fractional Fourier Transform (FRFT) Scheme Based on Closed Newton-Cotes Rules"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4296-2251","authenticated-orcid":false,"given":"Aubain","family":"Nzokem","sequence":"first","affiliation":[{"name":"Department of Mathematics & Statistics, York University, Toronto, ON M3J 1P3, Canada"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2641-0502","authenticated-orcid":false,"given":"Daniel","family":"Maposa","sequence":"additional","affiliation":[{"name":"Department of Statistics and Operations Research, University of Limpopo, Sovenga 0727, South Africa"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7485-3477","authenticated-orcid":false,"given":"Anna M.","family":"Seimela","sequence":"additional","affiliation":[{"name":"Department of Statistics and Operations Research, University of Limpopo, Sovenga 0727, South Africa"}]}],"member":"1968","published-online":{"date-parts":[[2025,7,20]]},"reference":[{"key":"ref_1","first-page":"48","article-title":"Comprehensive Survey on Fractional Fourier Transform","volume":"151","author":"Zhang","year":"2017","journal-title":"Fundam. 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