Smooth Selection for Infinite Sets

Whitney's extension problem asks the following: Given a compact set $E\subset\mathbb{R}^n$ and a function $f:E\to \mathbb{R}$, how can we tell whether there exists $F\in C^m(\mathbb{R}^n)$ such that $F|_E=f$? A 2006 theorem of Charles Fefferman answers this question in its full generality. In this paper, we establish a version of this theorem adapted for variants of the Whitney extension problem, including nonnegative extensions and the smooth selection problems. Among other things, we generalize the results of Fefferman-Israel-Luli (2016) to the setting of infinite sets.