Roots, Trace, and Extendability of Flat Nonnegative Smooth Functions

Fushuai Jiang

doi:10.1093/imrn/rnae151

Roots, Trace, and Extendability of Flat Nonnegative Smooth Functions

Fushuai Jiang^*

^*Corresponding author for this work

Research output: Journal Publications and Reviews › RGC 21 - Publication in refereed journal › peer-review

Abstract

Building on the univariate techniques developed by Ray and Schmidt-Hieber, we study the class F^s(Rⁿ) of multivariate nonnegative smooth functions that are sufficiently flat near their zeroes, which guarantees that F^r has Hölder differentiability rs whenever F ∈ F^s. We then construct a continuous Whitney extension map that recovers an F^s function from prescribed jets. Finally, we prove a Brudnyi–Shvartsman Finiteness Principle for the class F^s, thereby providing a necessary and sufficient condition for a nonnegative function defined on an arbitrary subset of Rⁿ to be F^s-extendable to all of Rⁿ. © The Author(s) 2024.

Original language	English
Pages (from-to)	12021-12043
Number of pages	23
Journal	International Mathematics Research Notices
Volume	2024
Issue number	17
Online published	5 Jul 2024
DOIs	https://doi.org/10.1093/imrn/rnae151
Publication status	Published - Sept 2024
Externally published	Yes

Bibliographical note

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© 2024 The Author(s). Published by Oxford University Press. All rights reserved.

Funding

I am grateful to my former PhD advisor Kevin Luli for introducing me to Whitney’s extension problems and for his valuable suggestions on this manuscript. I would like to thank Kolyan Ray and Johannes Schmidt-Hieber for their insightful comments on the multivariate counterpart of their original results. I would also like to thank the anonymous referees for their detailed suggestions of the manuscript. Part of this material is based upon work supported by the National Science Foundation under Grant No. DMS-1439786 while I was in residence at the Institute for Computational and Experimental Research in Mathematics in Providence, RI, during the fall 2022 semester.

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AB - 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. © The Author(s) 2024.

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M3 - RGC 21 - Publication in refereed journal

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SN - 1073-7928

VL - 2024

SP - 12021

EP - 12043

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

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ER -

Roots, Trace, and Extendability of Flat Nonnegative Smooth Functions

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