Roots, trace, and extendability of flat nonnegative smooth functions
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866909067334647808 |
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| author | Jiang, Fushuai |
| author_facet | Jiang, Fushuai |
| contents | Building on the univariate techniques developed by Ray and Schmidt-Hieber, we study the class $\mathcal{F}^s(\mathbb{R}^n)$ of multivariate nonnegative smooth functions that are sufficiently flat near their zeroes, which guarantees that $φ^r$ has Hölder differentiability $rs$ whenever $φ\in \mathcal{F}^s$. We then construct a continuous Whitney extension map that recovers an $\mathcal{F}^s$ function from prescribed jets. Finally, we prove a Brudnyi-Shvartsman Finiteness Principle for the class $\mathcal{F}^s$, thereby providing a necessary and sufficient condition for a nonnegative function defined on an arbitrary subset of $\mathbb{R}^n$ to be $\mathcal{F}^s$-extendable to all of $\mathbb{R}^n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_16183 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Roots, trace, and extendability of flat nonnegative smooth functions Jiang, Fushuai Functional Analysis Classical Analysis and ODEs Optimization and Control 26B35, 26B05, 41A29 Building on the univariate techniques developed by Ray and Schmidt-Hieber, we study the class $\mathcal{F}^s(\mathbb{R}^n)$ of multivariate nonnegative smooth functions that are sufficiently flat near their zeroes, which guarantees that $φ^r$ has Hölder differentiability $rs$ whenever $φ\in \mathcal{F}^s$. We then construct a continuous Whitney extension map that recovers an $\mathcal{F}^s$ function from prescribed jets. Finally, we prove a Brudnyi-Shvartsman Finiteness Principle for the class $\mathcal{F}^s$, thereby providing a necessary and sufficient condition for a nonnegative function defined on an arbitrary subset of $\mathbb{R}^n$ to be $\mathcal{F}^s$-extendable to all of $\mathbb{R}^n$. |
| title | Roots, trace, and extendability of flat nonnegative smooth functions |
| topic | Functional Analysis Classical Analysis and ODEs Optimization and Control 26B35, 26B05, 41A29 |
| url | https://arxiv.org/abs/2306.16183 |