We prove the computability of a version of Whitney Extension, when the input is suitably represented. More specifically, if <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$F \subseteq \mathbb {R}^n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>F</mi> <mo>⊆</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> is a closed set represented so that the distance function <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$x \mapsto d(x,F)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>↦</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>F</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> can be computed, and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$(f^{(\bar{k})})_{|\bar{k}| \le m}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mover accent="true"> <mrow> <mi>k</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">|</mo> <mover accent="true"> <mrow> <mi>k</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> <mo stretchy="false">|</mo> <mo>≤</mo> <mi>m</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> is a Whitney jet of order m on F, then we can compute <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$g \in C^{m}(\mathbb {R}^n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo>∈</mo> <msup> <mi>C</mi> <mi>m</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> such that g and its partial derivatives coincide on F with the corresponding functions of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$(f^{(\bar{k})})_{|\bar{k}| \le m}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mover accent="true"> <mrow> <mi>k</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">|</mo> <mover accent="true"> <mrow> <mi>k</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> <mo stretchy="false">|</mo> <mo>≤</mo> <mi>m</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>.

Computability of a whitney extension

期刊论文
发表日期：

2026年5月17日

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Andrea Brun,
Guido Gherardi,
Alberto Marcone

摘要

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