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Roots, trace, and extendability of flat nonnegative smooth functions

Fushuai Jiang

TL;DR

The work develops a multivariate theory for nonnegative smooth functions with flat behavior near zeros, introducing the class ${\mathcal{F}}^s(\Omega)$ and a flatness-based norm that guarantees Hölder regularity of roots. It then solves Whitney-extension problems for ${\mathcal{F}}^s$: (i) extending prescribed jets via a continuous Whitney map that preserves nonnegativity and bounds, and (ii) extending nonnegative functions from arbitrary closed sets by establishing a Finite-Set Principle with universal constants. The results connect to classical Whitney theory while handling nonnegativity and flatness, culminating in a shape-field framework that yields a practical criterion for extendability from arbitrary data. Collectively, the findings provide both theoretical foundations and constructive tools for extending nonnegative flat smooth functions in high dimensions, with potential applications in interpolation, optimization, and related numerical methods.

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 Hölder 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 $\mathbb{R}^n$.

Roots, trace, and extendability of flat nonnegative smooth functions

TL;DR

The work develops a multivariate theory for nonnegative smooth functions with flat behavior near zeros, introducing the class and a flatness-based norm that guarantees Hölder regularity of roots. It then solves Whitney-extension problems for : (i) extending prescribed jets via a continuous Whitney map that preserves nonnegativity and bounds, and (ii) extending nonnegative functions from arbitrary closed sets by establishing a Finite-Set Principle with universal constants. The results connect to classical Whitney theory while handling nonnegativity and flatness, culminating in a shape-field framework that yields a practical criterion for extendability from arbitrary data. Collectively, the findings provide both theoretical foundations and constructive tools for extending nonnegative flat smooth functions in high dimensions, with potential applications in interpolation, optimization, and related numerical methods.

Abstract

Building on the univariate techniques developed by Ray and Schmidt-Hieber, we study the class of multivariate nonnegative smooth functions that are sufficiently flat near their zeroes, which guarantees that has Hölder differentiability whenever . We then construct a continuous Whitney extension map that recovers an function from prescribed jets. Finally, we prove a Brudnyi-Shvartsman Finiteness Principle for the class , thereby providing a necessary and sufficient condition for a nonnegative function defined on an arbitrary subset of to be -extendable to all of .
Paper Structure (18 sections, 16 theorems, 127 equations)

This paper contains 18 sections, 16 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Definitions and notations
  3. Basic results on the flat norm
  4. Whitney jets
  5. A model bump function
  6. Extension of a single jet
  7. Case I: suppose $\delta \geq 1$.
  8. Case II: suppose $\delta < 1$.
  9. Whitney Extension Theorem for finite sets
  10. Case I: $F(x)\geq \delta^s\cdot M_2$.
  11. Case II: $F(x) < \delta^s\cdot M_2$
  12. Whitney Extension Theorem for arbitrary closed sets
  13. Finiteness Principles
  14. Shape Fields
  15. Proof of Theorem \ref{['thm:fp-arbitrary']}
  16. ...and 3 more sections

Key Result

Theorem 1.1

For $r \in (0,1]$, $s > 0$, a convex domain $\Omega\subset \mathbb{R}^n$, and all nonnegative $F \in {\mathcal{F}^s}(\Omega)$,

Theorems & Definitions (29)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: Finiteness Principle for ${\mathcal{F}^s}(\mathbb{R}^n)$
  • Proposition 3.1
  • Remark 3.1
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Definition 4.1
  • ...and 19 more