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arXiv · 1411.2451

On Smooth Whitney Extensions of almost isometries with small distortion, Interpolation and Alignment in $\Bbb R^D$-Part 1

S. B. Damelin·C. Fefferman
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Abstract

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 $\phi:E\to \mathbb R^D$ with $\phi$ a small distortion on $E$. How can one decide whether $\phi$ extends to a smooth small distortion $\Phi:\mathbb R^D\to \mathbb R^D$ which agrees with $\phi$ on $E$. We also ask how to decide if in addition $\Phi$ 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].

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S. B. Damelin, C. Fefferman. 2014-11-10. On Smooth Whitney Extensions of almost isometries with small distortion, Interpolation and Alignment in $\Bbb R^D$-Part 1. https://arxiv.org/abs/1411.2451

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