Computability of a Whitney Extension

TL;DR

This paper establishes the computability of Whitney Extension for functions defined on closed sets in Euclidean space, given a suitable representation that allows for the computation of the distance function.

Contribution

It proves that under specific representations, the Whitney Extension operator is computable for functions on closed sets in Euclidean space.

Findings

The extension function can be computed from the Whitney jet and the distance function.

The result applies to functions with derivatives up to order m on closed sets.

Computability is achieved under suitable representations of the input data.

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 $(f^{(\bar{k})})_{|\bar{k}| \le m}$.