Deep Univariate Polynomial and Conformal Approximation
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866915256010276864 |
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| author | Yeon, Kingsley |
| author_facet | Yeon, Kingsley |
| contents | A deep approximation is an approximating function defined by composing more than one layer of simple functions. We study deep approximations of functions of one variable using layers consisting of low-degree polynomials or simple conformal transformations. We show that deep approximations to $|x|$ on $[-1,1]$ achieve exponential convergence with respect to the degrees of freedom. Computational experiments suggest that a composite of two and three polynomial layers can give more accurate approximations than a single polynomial with the same number of coefficients. We also study the related problem of reducing the Runge phenomenon by composing polynomials with conformal transformations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_00698 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Deep Univariate Polynomial and Conformal Approximation Yeon, Kingsley Numerical Analysis A deep approximation is an approximating function defined by composing more than one layer of simple functions. We study deep approximations of functions of one variable using layers consisting of low-degree polynomials or simple conformal transformations. We show that deep approximations to $|x|$ on $[-1,1]$ achieve exponential convergence with respect to the degrees of freedom. Computational experiments suggest that a composite of two and three polynomial layers can give more accurate approximations than a single polynomial with the same number of coefficients. We also study the related problem of reducing the Runge phenomenon by composing polynomials with conformal transformations. |
| title | Deep Univariate Polynomial and Conformal Approximation |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2503.00698 |