Fourier Residual Networks Achieve Spectral Accuracy for Discontinuous Functions
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914530721792000 |
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| author | Davis, Owen Motamed, Mohammad Runborg, Olof |
| author_facet | Davis, Owen Motamed, Mohammad Runborg, Olof |
| contents | We present a constructive approximation framework for analyzing the expressive power of Fourier residual networks in approximating a broad class of one-dimensional functions. Our study covers both piecewise continuous functions -- including those with jump discontinuities in the function and its derivatives -- and fully smooth functions. We show that Fourier residual networks achieve spectral convergence without requiring periodicity or continuity, thereby overcoming key limitations of classical linear Fourier approximation and nonlinear methods, without being restricted to Barron-type function spaces. Our approach builds on classical techniques from approximation theory, including fixed-point iteration and Hermite interpolation by trigonometric polynomials. We support our theoretical results with numerical experiments based on both the constructed approximations and a randomized algorithm developed in our earlier work. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_03549 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Fourier Residual Networks Achieve Spectral Accuracy for Discontinuous Functions Davis, Owen Motamed, Mohammad Runborg, Olof Numerical Analysis Primary 41Axx, Secondary 68T07, 65T40 We present a constructive approximation framework for analyzing the expressive power of Fourier residual networks in approximating a broad class of one-dimensional functions. Our study covers both piecewise continuous functions -- including those with jump discontinuities in the function and its derivatives -- and fully smooth functions. We show that Fourier residual networks achieve spectral convergence without requiring periodicity or continuity, thereby overcoming key limitations of classical linear Fourier approximation and nonlinear methods, without being restricted to Barron-type function spaces. Our approach builds on classical techniques from approximation theory, including fixed-point iteration and Hermite interpolation by trigonometric polynomials. We support our theoretical results with numerical experiments based on both the constructed approximations and a randomized algorithm developed in our earlier work. |
| title | Fourier Residual Networks Achieve Spectral Accuracy for Discontinuous Functions |
| topic | Numerical Analysis Primary 41Axx, Secondary 68T07, 65T40 |
| url | https://arxiv.org/abs/2605.03549 |