Roots, Trace, and Extendability of Flat Nonnegative Smooth Functions

Abstract

Abstract Building on the univariate techniques developed by Ray and Schmidt-Hieber, we study the class $\mathcal{F}^{s}(\mathbb{R}^{n})$ of multivariate nonnegative smooth functions that are sufficiently flat near their zeroes, which guarantees that $F^{r}$ has Hölder differentiability $rs$ whenever $F \in \mathcal{F}^{s}$. We then construct a continuous Whitney extension map that recovers an $\mathcal{F}^{s}$ function from prescribed jets. Finally, we prove a Brudnyi–Shvartsman Finiteness Principle for the class $\mathcal{F}^{s}$, thereby providing a necessary and sufficient condition for a nonnegative function defined on an arbitrary subset of $\mathbb{R}^{n}$ to be $\mathcal{F}^{s}$-extendable to all of $\mathbb{R}^{n}$.