On Smooth Whitney Extensions of almost isometries with small distortion, Interpolation and Alignment in $\Bbb R^D$-Part 1
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| Format: | Preprint |
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2014
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| _version_ | 1866911783090913280 |
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| author | Damelin, S. B. Fefferman, C. |
| author_facet | Damelin, S. B. Fefferman, C. |
| contents | In this paper, we study the following problem: Let $D\geq 2$ and let $E\subset \mathbb R^D$ be finite satisfying certain conditions. Suppose that we are given a map $ϕ:E\to \mathbb R^D$ with $ϕ$ a small distortion on $E$. How can one decide whether $ϕ$ extends to a smooth small distortion $Φ:\mathbb R^D\to \mathbb R^D$ which agrees with $ϕ$ on $E$. We also ask how to decide if in addition $Φ$ can be approximated well by certain rigid and non-rigid motions from $\mathbb R^D\to \mathbb R^D$. Since $E$ is a finite set, this question is basic to interpolation and alignment of data in $\mathbb R^D$. The work in this paper appears in the research memoir [10]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1411_2451 |
| institution | arXiv |
| publishDate | 2014 |
| record_format | arxiv |
| spellingShingle | On Smooth Whitney Extensions of almost isometries with small distortion, Interpolation and Alignment in $\Bbb R^D$-Part 1 Damelin, S. B. Fefferman, C. Metric Geometry 58C25, 42B35, 94A08, 94C30, 41A05, 68Q25, 30E05, 26E10, 68Q17 In this paper, we study the following problem: Let $D\geq 2$ and let $E\subset \mathbb R^D$ be finite satisfying certain conditions. Suppose that we are given a map $ϕ:E\to \mathbb R^D$ with $ϕ$ a small distortion on $E$. How can one decide whether $ϕ$ extends to a smooth small distortion $Φ:\mathbb R^D\to \mathbb R^D$ which agrees with $ϕ$ on $E$. We also ask how to decide if in addition $Φ$ can be approximated well by certain rigid and non-rigid motions from $\mathbb R^D\to \mathbb R^D$. Since $E$ is a finite set, this question is basic to interpolation and alignment of data in $\mathbb R^D$. The work in this paper appears in the research memoir [10]. |
| title | On Smooth Whitney Extensions of almost isometries with small distortion, Interpolation and Alignment in $\Bbb R^D$-Part 1 |
| topic | Metric Geometry 58C25, 42B35, 94A08, 94C30, 41A05, 68Q25, 30E05, 26E10, 68Q17 |
| url | https://arxiv.org/abs/1411.2451 |