Roots, trace, and extendability of flat nonnegative smooth functions

TL;DR

This paper extends univariate techniques to multivariate nonnegative smooth functions, establishing their flatness near zeroes, constructing Whitney extension maps, and providing a finiteness principle for their extendability.

Contribution

It introduces a multivariate class of nonnegative smooth functions with flatness properties, constructs a Whitney extension map, and proves a finiteness principle for their extension.

Findings

Characterization of flatness near zeroes for multivariate functions

Construction of a continuous Whitney extension map

A necessary and sufficient condition for extendability of nonnegative functions

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 $\varphi^r$ has H\"older differentiability $rs$ whenever $\varphi \in \mathcal{F}^s$. 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 $\mathbb{R}^n$.