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On the Whitney Extension-Interpolation-Alignment problem for almost isometries with small distortion in $\Bbb R^D$

S. B Damelin, C. Fefferman

Abstract

In this paper, we study the following problem. Let $D\geq 2$, $S\subset \mathbb R^D$ be finite and let $φ:S\to \mathbb R^D$ with $φ$ a small distortion on $S$. We solve the Whitney extension-interpolation-alignment problem of how to understand when $φ$ can be extended to a function $Φ:\mathbb R^D\to \mathbb R^D$ which is a smooth small distortion on $\mathbb R^D$. The work in this paper appears in the research memoir [14]

On the Whitney Extension-Interpolation-Alignment problem for almost isometries with small distortion in $\Bbb R^D$

Abstract

In this paper, we study the following problem. Let , be finite and let with a small distortion on . We solve the Whitney extension-interpolation-alignment problem of how to understand when can be extended to a function which is a smooth small distortion on . The work in this paper appears in the research memoir [14]

Paper Structure

This paper contains 33 sections, 22 theorems, 79 equations.

Table of Contents

  1. Whitney's extension problem.
  2. Whitney's extension problem for small distortions in $\mathbb R^D$.
  3. Main Results: Whitney extensions-Interpolation-Alignment in $\mathbb R^D$.
  4. Three needed definitions.
  5. Five main results: Whitney extensions-interpolation and alignment in $\mathbb R^D$.
  6. Discussion of Main Results.
  7. Comparisons.
  8. The intriguing restriction in Theorem \ref{['t:lemmafive']} on the number of points being bounded by $n$, the dimension of the Euclidean space $\mathbb R^n$.
  9. The constants $\varepsilon$ and $\delta$ and how they relate to each other.
  10. Alignment of data in Euclidean space and the Procrustes problem.
  11. Labelled approximate alignment in $\mathbb R^n$ in the case where pairwise distances are distorted.
  12. Lojasiewicz's inequality and Proof of Theorem \ref{['t:lemmafour']}.
  13. $\varepsilon'$-distorted diffeomorphisms from $\mathbb R^n\to \mathbb R^n$; further properties and examples: Slow twists and Slides.
  14. Three useful properties of $\varepsilon'$-distorted diffeomorphisms from $\mathbb R^n\to \mathbb R^n$.
  15. Slow twists.
  16. ...and 18 more sections

Key Result

Theorem 2.3

Let $S\subset \mathbb R^D$ be finite. There exists positive constants $c_K$, $C'_K$, $C"_K$ depending only on $D$ and $K$ such that the following holds: Set $\eta=\exp(-C'_K/\varepsilon)$ and $\delta=\exp(-C"_K/\varepsilon)$ with $0<\varepsilon<c_K$. Let $\phi:S\to \mathbb R^D$ satisfy Then if $\phi$ has no negative $\eta$ block, there exists a proper $\varepsilon$-distorted diffeomorphism $\Phi:

Theorems & Definitions (27)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Theorem 3.8
  • Theorem 3.9
  • Theorem 3.10
  • ...and 17 more