Inducing Riesz and orthonormal bases in $L^2$ via composition operators
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866918185136029696 |
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| author | Saleh, Yahya Iske, Armin |
| author_facet | Saleh, Yahya Iske, Armin |
| contents | Let $C_h$ be a composition operator mapping $L^2(Ω_1)$ into $L^2(Ω_2)$ for some open sets $Ω_1, Ω_2 \subseteq \mathbb{R}^n$. We characterize the mappings $h$ that transform Riesz bases of $L^2(Ω_1)$ into Riesz bases of $L^2(Ω_2)$. Restricting our analysis to differentiable mappings, we demonstrate that mappings $h$ that preserve Riesz bases have Jacobian determinants that are bounded away from zero and infinity. We discuss implications of these results for approximation theory, highlighting the potential of using bijective neural networks to construct Riesz bases with favorable approximation properties. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_18613 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Inducing Riesz and orthonormal bases in $L^2$ via composition operators Saleh, Yahya Iske, Armin Functional Analysis Machine Learning Numerical Analysis 47B33, 42C15 Let $C_h$ be a composition operator mapping $L^2(Ω_1)$ into $L^2(Ω_2)$ for some open sets $Ω_1, Ω_2 \subseteq \mathbb{R}^n$. We characterize the mappings $h$ that transform Riesz bases of $L^2(Ω_1)$ into Riesz bases of $L^2(Ω_2)$. Restricting our analysis to differentiable mappings, we demonstrate that mappings $h$ that preserve Riesz bases have Jacobian determinants that are bounded away from zero and infinity. We discuss implications of these results for approximation theory, highlighting the potential of using bijective neural networks to construct Riesz bases with favorable approximation properties. |
| title | Inducing Riesz and orthonormal bases in $L^2$ via composition operators |
| topic | Functional Analysis Machine Learning Numerical Analysis 47B33, 42C15 |
| url | https://arxiv.org/abs/2406.18613 |