Reconstruction of Piecewise Smooth Functions Based on Fourier Extension
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–namely boundary-induced artifacts, instability in edge detection, and loss of accuracy near...
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| Language: | English |
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MDPI AG
2025-06-01
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| Online Access: | https://www.mdpi.com/2075-1680/14/7/489 |
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| _version_ | 1850078430507302912 |
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| author | Xusheng Li Zhenyu Zhao Xianzheng Jia |
| author_facet | Xusheng Li Zhenyu Zhao Xianzheng Jia |
| author_sort | Xusheng Li |
| collection | DOAJ |
| description | 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–namely boundary-induced artifacts, instability in edge detection, and loss of accuracy near discontinuities–the 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 (∼10<sup>−14</sup>–10<sup>−15</sup>) with significantly improved stability and performance over traditional global methods. |
| format | Article |
| id | doaj-art-614e89586acc40e683ddb3518eef09e1 |
| institution | DOAJ |
| issn | 2075-1680 |
| language | English |
| publishDate | 2025-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj-art-614e89586acc40e683ddb3518eef09e12025-08-20T02:45:33ZengMDPI AGAxioms2075-16802025-06-0114748910.3390/axioms14070489Reconstruction of Piecewise Smooth Functions Based on Fourier ExtensionXusheng Li0Zhenyu Zhao1Xianzheng Jia2School of Mathematics and Statistics, Shandong University of Technology, Zibo 255049, ChinaSchool of Mathematics and Statistics, Shandong University of Technology, Zibo 255049, ChinaSchool of Mathematics and Statistics, Shandong University of Technology, Zibo 255049, ChinaThis 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–namely boundary-induced artifacts, instability in edge detection, and loss of accuracy near discontinuities–the 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 (∼10<sup>−14</sup>–10<sup>−15</sup>) with significantly improved stability and performance over traditional global methods.https://www.mdpi.com/2075-1680/14/7/489Fourier seriesFourier extensionGibbs phenomenonpiecewise smooth functions |
| spellingShingle | Xusheng Li Zhenyu Zhao Xianzheng Jia Reconstruction of Piecewise Smooth Functions Based on Fourier Extension Axioms Fourier series Fourier extension Gibbs phenomenon piecewise smooth functions |
| title | Reconstruction of Piecewise Smooth Functions Based on Fourier Extension |
| title_full | Reconstruction of Piecewise Smooth Functions Based on Fourier Extension |
| title_fullStr | Reconstruction of Piecewise Smooth Functions Based on Fourier Extension |
| title_full_unstemmed | Reconstruction of Piecewise Smooth Functions Based on Fourier Extension |
| title_short | Reconstruction of Piecewise Smooth Functions Based on Fourier Extension |
| title_sort | reconstruction of piecewise smooth functions based on fourier extension |
| topic | Fourier series Fourier extension Gibbs phenomenon piecewise smooth functions |
| url | https://www.mdpi.com/2075-1680/14/7/489 |
| work_keys_str_mv | AT xushengli reconstructionofpiecewisesmoothfunctionsbasedonfourierextension AT zhenyuzhao reconstructionofpiecewisesmoothfunctionsbasedonfourierextension AT xianzhengjia reconstructionofpiecewisesmoothfunctionsbasedonfourierextension |