{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:26:12Z","timestamp":1760059572554,"version":"build-2065373602"},"reference-count":22,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2025,6,23]],"date-time":"2025-06-23T00:00:00Z","timestamp":1750636800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Shandong University of Technology Doctoral Start-up Fund","award":["419047"],"award-info":[{"award-number":["419047"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper proposes a hierarchical Fourier extension framework for the accurate reconstruction of piecewise smooth functions with mixed-order singularities. To address key challenges in spectral approximation\u2013namely boundary-induced artifacts, instability in edge detection, and loss of accuracy near discontinuities\u2013the method integrates three main components: (1) boundary-focused Fourier extensions that isolate endpoint effects while preserving internal structures; (2) a multi-stage edge detection strategy combining spectral mollifiers and coordinate transformations to identify discontinuities in function values and their derivatives; (3) adaptive domain partitioning followed by localized Fourier extensions to retain spectral accuracy on smooth segments. Numerical results demonstrate near machine-precision accuracy (\u223c10\u221214\u201310\u221215) with significantly improved stability and performance over traditional global methods.<\/jats:p>","DOI":"10.3390\/axioms14070489","type":"journal-article","created":{"date-parts":[[2025,6,23]],"date-time":"2025-06-23T07:42:34Z","timestamp":1750664554000},"page":"489","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Reconstruction of Piecewise Smooth Functions Based on Fourier Extension"],"prefix":"10.3390","volume":"14","author":[{"given":"Xusheng","family":"Li","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Shandong University of Technology, Zibo 255049, China"}]},{"given":"Zhenyu","family":"Zhao","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Shandong University of Technology, Zibo 255049, China"}]},{"given":"Xianzheng","family":"Jia","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Shandong University of Technology, Zibo 255049, China"}]}],"member":"1968","published-online":{"date-parts":[[2025,6,23]]},"reference":[{"key":"ref_1","unstructured":"Cockburn, B., Shu, C.W., Johnson, C., Tadmor, E., and Shu, C.W. 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