A universal approximation theorem and its applications to vector lattice theory
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910080403767296 |
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| author | Bilokopytov, Eugene Xanthos, Foivos |
| author_facet | Bilokopytov, Eugene Xanthos, Foivos |
| contents | A classical result in approximation theory states that for any continuous function \( φ: \mathbb{R} \to \mathbb{R} \), the set \( \operatorname{span}\{φ\circ g : g \in \operatorname{Aff}(\mathbb{R})\} \) is dense in \( \mathcal{C}(\mathbb{R}) \) if and only if \( φ\) is not a polynomial. In this note, we present infinite dimensional variants of this result. These extensions apply to neural network architectures and improve the main density result obtained in \cite{BDG23}. We also discuss applications and related approximation results in vector lattices, improving and complementing results from \cite{AT:17, bhp,BT:24}. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_20219 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A universal approximation theorem and its applications to vector lattice theory Bilokopytov, Eugene Xanthos, Foivos Functional Analysis 41A65, 46A19, 46A40, 46B42, 46E25, 46E40, 68T07 A classical result in approximation theory states that for any continuous function \( φ: \mathbb{R} \to \mathbb{R} \), the set \( \operatorname{span}\{φ\circ g : g \in \operatorname{Aff}(\mathbb{R})\} \) is dense in \( \mathcal{C}(\mathbb{R}) \) if and only if \( φ\) is not a polynomial. In this note, we present infinite dimensional variants of this result. These extensions apply to neural network architectures and improve the main density result obtained in \cite{BDG23}. We also discuss applications and related approximation results in vector lattices, improving and complementing results from \cite{AT:17, bhp,BT:24}. |
| title | A universal approximation theorem and its applications to vector lattice theory |
| topic | Functional Analysis 41A65, 46A19, 46A40, 46B42, 46E25, 46E40, 68T07 |
| url | https://arxiv.org/abs/2507.20219 |