Computability of a Whitney Extension

2025/07/02 by Brun, Andrea, Gherardi, Guido, Marcone, Alberto
#03D78 #26E10 #54C20 #54C30 #58C25 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.2507.02113

Abstract

We prove the computability of a version of Whitney Extension, when the input is suitably represented. More specifically, if F ⊆ ℝⁿ is a closed set represented so that the distance function x ↦ d(x,F) can be computed, and (f^(k))_|k| ≤ m is a Whitney jet of order m on F, then we can compute g ∈ C^m(ℝⁿ) such that g and its partial derivatives coincide on F with the corresponding functions of (f^(k))_|k| ≤ m.

Citations

• A Whitney Extension Problem for Manifolds
• Open sets in computability theory and Reverse Mathematics
• The reverse mathematics of the Tietze extension theorem
• Analytic extensions of differentiable functions defined in closed sets

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• Whitney extension operators without loss of derivatives
• Mityagin's Extension Problem. Progress Report
• A coordinate-free proof of the finiteness principle for the Whitney extension problem
• The Whitney extension problem and Lipschitz selections of set-valued mappings in jet-spaces