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

Abstract

In this paper, we study the following problem: Let $D\ge 2$ and let $E\subset {\mathbb{R}}^D$ be finite satisfying certain conditions. Suppose that we are given a map $\varphi :E\to {\mathbb{R}}^D$ with $\varphi$ a small distortion on $E$. How can one decide whether $\varphi$ extends to a smooth small distortion $\Phi :{\mathbb{R}}^D\to {\mathbb{R}}^D$ which agrees with $\varphi$ 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].

arXiv commentThe material for this paper appears in the research memoir: S. B. Damelin, Near extensions and Alignment of data in ${\mathbb{R}}^n$: Whitney extensions of smooth near isometries, shortest paths, equidistribution, clustering and non-rigid alignment of data in Euclidean space, John Wiley & Sons, November 2023