Roots, Trace, and Extendability of Flat Nonnegative Smooth Functions

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