nLab Whitney extension theorem

Idea

A statement about sufficient data for extensions of a smooth function from a compact subset to an open neighbourhood.

Related concepts

extension theorems	continuous functions	smooth functions
plain functions	Tietze extension theorem	Whitney extension theorem
equivariant functions	equivariant Tietze extension theorem

• Hadamard lemma,

• Borel's theorem

• Tietze extension theorem

• Taimanov theorem

• Steenrod-Wockel approximation theorem

• embedding of smooth manifolds into formal duals of R-algebras

• smooth Serre-Swan theorem

• derivations of smooth functions are vector fields

References

The original article is

• Hassler Whitney, Analytic extensions of differentiable functions defined in closed sets, 1933 (pdf)

Textbook accounts include

• Lars Hörmander, theore 2.3.6 of The analysis of linear partial differential operators, vol. I, Springer 1983, 1990

Enhancement to a linear splitting of restriction maps on Fréchet spaces of sections with compact support of vector bundles:

• David Roberts, Alexander Schmeding, Extending Whitney’s extension theorem: nonlinear function spaces (arXiv:1801.04126)

This is then used to show the restriction map to (suitable) regular closed subsets is a submersion of mapping spaces (with maps valued in an arbitrary manifold).

See also

• Wikipedia, Whitney extension theorem

• eom, Whitney extension theorem

• Armin Rainer, Ultradifferentiable extension theorems: a survey, Expositiones Mathematicae (2021) [arXiv:2107.01061, doi:10.1016/j.exmath.2021.12.001]