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 ${\mathbb{R}}^n$

Abstract

In this announcement we consider the following problem. Let $n,m\ge 1$, $U\subset {\mathbb{R}}^n$ open. In this paper we provide a sharp solution to the following Whitney distortion extension problems: (a) Let $\varphi :U\to {\mathbb{R}}^n$ be a $C^m$ map. If $E\subset U$ is compact (with some geometry) and the restriction of $\varphi$ 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 $\Phi :{\mathbb{R}}^n\to {\mathbb{R}}^n$ with small distortion which agrees with $\varphi$ in a neighborhood of $E$ and a Euclidean motion $A:{\mathbb{R}}^n\to {\mathbb{R}}^n$ away from $E$. (b) Let $\varphi :U\to {\mathbb{R}}^n$ be $C^{\infty }$ map. If $E\subset U$ is compact (with some geometry) and the restriction of $\varphi$ 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 $\Phi :{\mathbb{R}}^n\to {\mathbb{R}}^n$ with small distortion which agrees with $\varphi$ 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].

The 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. arXiv admin note: substantial text overlap with arXiv:1411.2468

References in corpus (2)

• Cramér-Rao bounds for synchronization of rotations
• Fourier-Bessel rotational invariant eigenimages