A coordinate-free proof of the finiteness principle for the Whitney extension problem

2018/08/20 by Jacob Carruth, Abraham Frei-Pearson, Carruth, Jacob +5
Mathematics · #42B99 #Advanced Topology and Set Theory #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical and Theoretical Analysis

paper · pdf · doi:10.48550/arxiv.1808.06658

openalex publication_date 2018/08/20 · openalex created_date 2018/08/31 · openalex updated_date 2026/07/28

Abstract

We present a coordinate-free version of Fefferman's solution of Whitney's extension problem in the space C^m-1,1(ℝⁿ). While the original argument relies on an elaborate induction on collections of partial derivatives, our proof uses the language of ideals and translation-invariant subspaces in the ring of polynomials. We emphasize the role of compactness in the proof, first in the familiar sense of topological compactness, but also in the sense of finiteness theorems arising in logic and semialgebraic geometry.

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• The Whitney extension theorem in high dimensions
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• Whitney extensions and orthonormal expansions
• Smooth Selection for Infinite Sets
• Computability of a Whitney Extension