Extension operators for smooth functions on compact subsets of the reals

TL;DR

This paper establishes conditions under which smooth functions defined on compact subsets of the real line can be extended to the entire real line via continuous linear operators, addressing specific examples previously studied.

Contribution

It provides necessary and sufficient conditions for the existence of continuous linear extension operators for smooth functions on compact subsets of the real line.

Findings

Characterization of sets admitting extension operators

Application to sets with accumulation points at zero

Extension of previous examples by Fefferman, Ricci, and Vogt

Abstract

We introduce sufficient as well as necessary conditions for a compact set $K$ such that there is a continuous linear extension operator from the space of restrictions $C^\infty(K)=\lbrace F|_K: F\in C^\infty(\mathbb R)\rbrace$ to $C^\infty(\mathbb R)$. This allows us to deal with examples of the form $K=\lbrace a_n:n\in\mathbb N\rbrace \cup \lbrace 0\rbrace$ for $a_n\to 0$ previously considered by Fefferman and Ricci as well as Vogt.