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

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].