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Computability of a Whitney Extension

Andrea Brun, Guido Gherardi, Alberto Marcone

Abstract

We prove the computability of a version of Whitney Extension, when the input is suitably represented. More specifically, if $F \subseteq \mathbb{R}^n$ is a closed set represented so that the distance function $x \mapsto d(x,F)$ can be computed, and $(f^{(\bar{k})})_{|\bar{k}| \le m}$ is a Whitney jet of order $m$ on $F$, then we can compute $g \in C^{m}(\mathbb{R}^n)$ such that $g$ and its partial derivatives coincide on $F$ with the corresponding functions of $(f^{(\bar{k})})_{|\bar{k}| \le m}$.

Computability of a Whitney Extension

Abstract

We prove the computability of a version of Whitney Extension, when the input is suitably represented. More specifically, if is a closed set represented so that the distance function can be computed, and is a Whitney jet of order on , then we can compute such that and its partial derivatives coincide on with the corresponding functions of .

Paper Structure

This paper contains 12 sections, 33 theorems, 137 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Computable Analysis
  3. Represented Spaces
  4. Computability of multi-valued functions between represented spaces
  5. Representation of computable metric spaces
  6. Representation of continuous functions
  7. Decomposition of open sets into cubes
  8. Partition of unity
  9. First Computable Whitney Extension Theorem
  10. Computable Whitney Extension Theorem
  11. Open questions
  12. Proof of Proposition \ref{['Prop_Derivativesvarphi']}

Key Result

Theorem 2.8

If $X$ and $Y$ are $T_0$ second countable spaces with admissible representations $\delta$ and $\gamma$, respectively, then $f: \subseteq X \to Y$ is $(\delta, \gamma)$-realizer continuous if and only if it is topologically continuous. $\blacktriangleleft$$\blacktriangleleft$

Figures (3)

  • Figure 1: The covering of cubes
  • Figure 2:
  • Figure 3:

Theorems & Definitions (80)

  • Definition 2.1: Type-$2$ Turing machine
  • Definition 2.2: Represented space
  • Definition 2.3
  • Definition 2.4: Realizer
  • Definition 2.5: Computable and realizer continuous problems
  • Definition 2.6
  • Definition 2.7: Admissible representation
  • Theorem 2.8: weihrauch:2000
  • Definition 2.9: Computable metric space
  • Definition 2.10
  • ...and 70 more