The Whitney extension theorem in high dimensions

TL;DR

This paper presents a modified Whitney extension theorem applicable in high-dimensional spaces, demonstrating that the extension operator's norm grows polynomially with dimension for fixed smoothness level.

Contribution

It introduces a variant of the Whitney extension theorem with polynomial growth of the extension operator's norm in high dimensions.

Findings

Extension operator norm grows polynomially with dimension

Applicable to $ ext{C}^m( ext{R}^n)$ functions in high dimensions

Provides a dimensionally controlled extension method

Abstract

We prove a variant of the standard Whitney extension theorem for $\mathcal C^m(\mathbb R^n)$, in which the norm of the extension operator has polynomial growth in $n$ for fixed $m$.