Article
| Title: | On extending ${\rm C}^{k}$ functions from an open set to $\mathbb R$ with applications (English) |
| Author: | Burgess, Walter D. |
| Author: | Raphael, Robert M. |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 73 |
| Issue: | 2 |
| Year: | 2023 |
| Pages: | 487-498 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | For $k\in {\mathbb N} \cup \{\infty \}$ and $U$ open in $ {\mathbb R}$, let ${\rm C}^{k}(U)$ be the ring of real valued functions on $U$ with the first $k$ derivatives continuous. It is shown that for $f\in {\rm C}^{k}(U)$ there is $g\in {\rm C}^{\infty } ({\mathbb R})$ with $U\subseteq {\rm coz} g$ and $h\in {\rm C}^{k} ({\mathbb R})$ with $fg|_U=h|_U$. The function $f$ and its $k$ derivatives are not assumed to be bounded on $U$. The function $g$ is constructed using splines based on the Mollifier function. Some consequences about the ring ${\rm C}^{k} ({\mathbb R})$ are deduced from this, in particular that ${\rm Q}_{\rm cl} ({\rm C}^{k} ({\mathbb R})) = {\rm Q}({\rm C}^{k} ({\mathbb R}))$. (English) |
| Keyword: | ${\rm C}^k$ function |
| Keyword: | spline |
| Keyword: | ring of quotient |
| Keyword: | Mollifier function |
| MSC: | 13B30 |
| MSC: | 26A24 |
| MSC: | 54C30 |
| idZBL: | Zbl 07729519 |
| idMR: | MR4586906 |
| DOI: | 10.21136/CMJ.2023.0445-21 |
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| Date available: | 2023-05-04T17:46:43Z |
| Last updated: | 2023-09-13 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/151669 |
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