Some examples of $C^\infty$ extension by linear operators

Abstract

For two kinds of sets $X$ in $\mathbb{R}^n$, we prove the existence of linear continuous operators extending $C^\infty$ functions on $X$ to $C^\infty$ functions on $\mathbb{R}^n$. The sets we consider are: (a) sequences of points in the real line converging to 0 at a polynomial rate, (b) flag-shaped sets in the plane, which are unions of half-lines with slopes as in (a).