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arxiv: 2507.02113 · v2 · submitted 2025-07-02 · 🧮 math.LO · math.CA

Computability of a Whitney Extension

Andrea Brun , Guido Gherardi , Alberto Marcone This is my paper

Pith reviewed 2026-05-19 06:01 UTC · model grok-4.3

classification 🧮 math.LO math.CA
keywords computable analysisWhitney extensionclosed setsC^m functionsjet extensionseffective mathematicsdistance functions
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The pith

A Whitney jet of order m on a closed set F extends to a computable C^m function on R^n when the distance to F is computable.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that the Whitney extension theorem becomes effective under representations that make the distance function to a closed set computable. Given such an F in R^n and a Whitney jet of partial derivatives up to order m supplied in a compatible way, there is a procedure that outputs a global function g whose derivatives match the jet exactly on F. This matters because it turns a classical existence result into a constructive one, so that smooth extensions can be built algorithmically from data on sets that may be irregular or fractal. The result applies uniformly in any dimension and for any finite order of smoothness.

Core claim

If F is a closed subset of R^n presented so that the distance function x maps to d(x,F) is computable, and if a Whitney jet (f^(k)) for |k| <= m is given under a representation that permits effective manipulation, then one can compute a function g belonging to C^m(R^n) whose partial derivatives up to order m coincide with those of the jet on the set F.

What carries the argument

The effective Whitney extension operator that uses the computable distance to F to construct a global C^m function by blending the jet data away from the set.

If this is right

  • Smooth extensions of jets on computable closed sets such as algebraic curves or self-similar fractals become algorithmic.
  • Problems of C^m interpolation and approximation on closed domains can be solved by effective procedures rather than non-constructive existence proofs.
  • The same method yields computable extensions of any finite order m, so higher smoothness can be achieved without changing the representation of the input.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The result opens a route to effective versions of other classical extension theorems once suitable computable representations of the underlying sets are fixed.
  • Numerical schemes for differential equations on domains defined by computable distance functions could incorporate these extensions as building blocks.
  • Adaptation of the distance-based blending technique to manifolds or stratified spaces would require only corresponding changes in the representation of the base set.

Load-bearing premise

The closed set F must be represented so its distance function is computable and the Whitney jet must be presented in a form that lets the extension procedure be carried out by a computable process.

What would settle it

Exhibit a closed set F with a computable distance function together with a Whitney jet on F for which every candidate extension either fails to be C^m or fails to match the jet on F, or for which no computable extension exists at all.

Figures

Figures reproduced from arXiv: 2507.02113 by Alberto Marcone, Andrea Brun, Guido Gherardi.

Figure 1
Figure 1. Figure 1: The covering of cubes It remains then to show that F belongs to CUBE(F). As already seen, (C2) is obvious from the construction. Since our definition of F uses rational approxima￾tions and dense sets we need to prove (C1) explicitly, while Stein can take it for granted. Proposition 3.5. The family of cubes F covers the complement of F, that is [ Q∈F Q = R n \ F. Proof. If we prove that each point in R n \ … view at source ↗
Figure 2
Figure 2. Figure 2: By Proposition 3.11, we have 1 − ε 2 diam(Q) ≤ d(Q ∗ , F) < 3δ, and since d(Q∗ , F) < d(Q, F), using Propositions 3.6 and 3.10 we obtain δ < d(Q ∗ , F) + diam(Q ∗ ) < d(Q, F) + (1 + ε) diam(Q) < (6 + ε) diam(Q). Therefore we have upper and lower bounds for the diameter of the cubes Q that belong to Fex, i.e. δ 6 + ε < diam(Q) < 6δ 1 − ε . Now, using Proposition 3.10 we have δ > d(x, Q∗ ) ≥ d(x, Q) − ε 2 di… view at source ↗
Figure 3
Figure 3. Figure 3: 4. Partition of unity In this section, using the families of cubes F and F ∗ (defined respectively in (6) and (9)), we define a collection of C∞ computable functions which constitutes a partition of unity on R n \ F. Moreover all partial derivatives of these functions are computable. Let Q0 ⊆ R n be the cube with center (0, ..., 0) and edges parallel to the axes of length 1. We need to define a computable … view at source ↗
read the original abstract

We prove the computability of a version of Whitney Extension, when the input is suitably represented. More specifically, if $F \subseteq \mathbb{R}^n$ is a closed set represented so that the distance function $x \mapsto d(x,F)$ can be computed, and $(f^{(\bar{k})})_{|\bar{k}| \le m}$ is a Whitney jet of order $m$ on $F$, then we can compute $g \in C^{m}(\mathbb{R}^n)$ such that $g$ and its partial derivatives coincide on $F$ with the corresponding functions of $(f^{(\bar{k})})_{|\bar{k}| \le m}$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper proves a computable version of the Whitney extension theorem: if F ⊆ ℝ^n is closed and represented so that the distance function x ↦ d(x,F) is computable, and if (f^(k̄))_{|k̄|≤m} is a Whitney jet of order m on F supplied under a representation making the extension effective, then a function g ∈ C^m(ℝ^n) can be computed such that g and its partial derivatives up to order m agree with the jet on F.

Significance. If the result holds, it supplies an effective counterpart to the classical Whitney theorem within computable analysis. The work shows that the standard partition-of-unity or summation construction can be rendered computable once representations for the distance function and the jet are fixed appropriately, thereby contributing a concrete algorithmic bridge between classical real analysis and computability theory.

major comments (1)
  1. [§3] §3 (Main construction): the proof that the summed extension g is computable from the given representations of d(·,F) and the jet must be checked for the precise modulus of continuity or approximation rate used to truncate the sum; without an explicit effective bound on the remainder, the claim that g is computable remains formally incomplete.
minor comments (2)
  1. [§2] The precise definition of the representation for the Whitney jet (e.g., how the functions f^(k̄) and the Whitney remainder estimates are encoded as computable objects) should be stated explicitly in §2, with a short comparison to standard representations in computable analysis.
  2. [Throughout] Notation for multi-indices k̄ and the order m is used throughout; a single displayed definition of the jet space and the Whitney condition would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the recommendation for minor revision. The observation regarding the explicit effective bound in the main construction is well taken, and we address it directly below.

read point-by-point responses

  1. Referee: [§3] §3 (Main construction): the proof that the summed extension g is computable from the given representations of d(·,F) and the jet must be checked for the precise modulus of continuity or approximation rate used to truncate the sum; without an explicit effective bound on the remainder, the claim that g is computable remains formally incomplete.

    Authors: We agree that an explicit effective bound on the remainder is required for a fully rigorous computability argument. The construction in §3 uses the computable distance function and the given representation of the Whitney jet to control the tail of the sum via the standard Whitney remainder estimates. In the revised manuscript we will insert a new lemma that extracts a computable modulus of continuity for the remainder directly from these representations, thereby making the truncation effective and completing the proof that g is computable. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper delivers a direct constructive proof that a Whitney jet of order m on a closed set F, when F is represented so its distance function is computable and the jet is given under a compatible representation, yields a computable C^m extension. The argument proceeds by making the classical Whitney construction effective via the given representations, without any reduction of the target computability statement to a fitted parameter, a self-referential definition, or a load-bearing self-citation whose content is presupposed rather than independently verified. The derivation remains self-contained against the stated assumptions on representations.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on the classical Whitney extension theorem and standard definitions of computability for real-valued functions and sets via representations.

axioms (2)
  • standard math The classical Whitney extension theorem holds for closed sets in R^n.
    The paper proves a computable version of this known theorem.
  • domain assumption Real numbers, functions, and sets are represented so that computability means the existence of a Turing machine producing approximations to any precision.
    This is the standard framework in computable analysis used throughout the claim.

pith-pipeline@v0.9.0 · 5633 in / 1350 out tokens · 65887 ms · 2026-05-19T06:01:39.055348+00:00 · methodology

discussion (0)

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Reference graph

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