Extension of Whitney jets of controlled growth

TL;DR

This paper extends Whitney's theorem in the ultradifferentiable setting, providing explicit methods to extend Whitney jets with controlled growth from compact subsets to the whole space.

Contribution

It introduces a method to explicitly determine the class of Whitney jets that can be extended while preserving ultradifferentiability constraints.

Findings

Extension of Whitney jets with growth control is possible in the ultradifferentiable Roumieu setting.

Explicit computation of the extension class from the original class is achieved.

Results generalize previous theorems to arbitrary compact subsets in bR^n.

Abstract

We revisit Whitney's extension theorem in the ultradifferentiable Roumieu setting. Based on the description of ultradifferentiable classes by weight matrices, we extend results on how growth constraints on Whitney jets on arbitrary compact subsets in $\mathbb R^n$ are preserved by their extensions to $\mathbb R^n$. More precisely, for any admissible class $\mathcal C$ of ultradifferentiable functions on $\mathbb R^n$ we determine a class $\mathcal C'$ such that all ultradifferentiable Whitney jets of class $\mathcal C'$ on arbitrary compact subsets admit extensions in $\mathcal C$. The class $\mathcal C'$ can be explicitly computed from $\mathcal C$.