On the Whitney distortion extension problem for $C^m(\mathbb R^n)$ and $C^{\infty}(\mathbb R^n)$ and its applications to interpolation and alignment of data in $\mathbb R^n$
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arXiv
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2015
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| _version_ | 1866911783097204736 |
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| author | Damelin, S. B Fefferman, C. |
| author_facet | Damelin, S. B Fefferman, C. |
| contents | In this announcement we consider the following problem. Let $n,m\geq 1$, $U\subset\mathbb R^n$ open. In this paper we provide a sharp solution to the following Whitney distortion extension problems: (a) Let $ϕ:U\to \mathbb R^n$ be a $C^m$ map. If $E\subset U$ is compact (with some geometry) and the restriction of $ϕ$ to $E$ is an almost isometry with small distortion, how to decide when there exists a $C^m(\mathbb R^n)$ one-to-one and onto almost isometry $Φ:\mathbb R^n\to \mathbb R^n$ with small distortion which agrees with $ϕ$ in a neighborhood of $E$ and a Euclidean motion $A:\mathbb R^n\to \mathbb R^n$ away from $E$. (b) Let $ϕ:U\to \mathbb R^n$ be $C^{\infty}$ map. If $E\subset U$ is compact (with some geometry) and the restriction of $ϕ$ to $E$ is an almost isometry with small distortion, how to decide when there exists a $C^{\infty}(\mathbb R^n)$ one-to-one and onto almost isometry $Φ:\mathbb R^n\to \mathbb R^n$ with small distortion which agrees with $ϕ$ in a neighborhood of $E$ and a Euclidean motion $A:\mathbb R^n\to \mathbb R^n$ away from $E$. Our results complement those of [14,15,20] where there, $E$ is a finite set. In this case, the problem above is also a problem of interpolation and alignment of data in $\mathbb R^n$. The material in this paper appears in the memoir [14]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1505_06950 |
| institution | arXiv |
| publishDate | 2015 |
| record_format | arxiv |
| spellingShingle | On the Whitney distortion extension problem for $C^m(\mathbb R^n)$ and $C^{\infty}(\mathbb R^n)$ and its applications to interpolation and alignment of data in $\mathbb R^n$ Damelin, S. B Fefferman, C. Classical Analysis and ODEs 58C25, 42B35, 94A08, 94C30, 41A05, 68Q25, 30E05, 26E10, 68Q17 In this announcement we consider the following problem. Let $n,m\geq 1$, $U\subset\mathbb R^n$ open. In this paper we provide a sharp solution to the following Whitney distortion extension problems: (a) Let $ϕ:U\to \mathbb R^n$ be a $C^m$ map. If $E\subset U$ is compact (with some geometry) and the restriction of $ϕ$ to $E$ is an almost isometry with small distortion, how to decide when there exists a $C^m(\mathbb R^n)$ one-to-one and onto almost isometry $Φ:\mathbb R^n\to \mathbb R^n$ with small distortion which agrees with $ϕ$ in a neighborhood of $E$ and a Euclidean motion $A:\mathbb R^n\to \mathbb R^n$ away from $E$. (b) Let $ϕ:U\to \mathbb R^n$ be $C^{\infty}$ map. If $E\subset U$ is compact (with some geometry) and the restriction of $ϕ$ to $E$ is an almost isometry with small distortion, how to decide when there exists a $C^{\infty}(\mathbb R^n)$ one-to-one and onto almost isometry $Φ:\mathbb R^n\to \mathbb R^n$ with small distortion which agrees with $ϕ$ in a neighborhood of $E$ and a Euclidean motion $A:\mathbb R^n\to \mathbb R^n$ away from $E$. Our results complement those of [14,15,20] where there, $E$ is a finite set. In this case, the problem above is also a problem of interpolation and alignment of data in $\mathbb R^n$. The material in this paper appears in the memoir [14]. |
| title | On the Whitney distortion extension problem for $C^m(\mathbb R^n)$ and $C^{\infty}(\mathbb R^n)$ and its applications to interpolation and alignment of data in $\mathbb R^n$ |
| topic | Classical Analysis and ODEs 58C25, 42B35, 94A08, 94C30, 41A05, 68Q25, 30E05, 26E10, 68Q17 |
| url | https://arxiv.org/abs/1505.06950 |