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 articles:

• Hassler Whitney: Analytic extensions of differentiable functions defined in closed sets, Transactions of the American Mathematical Society 36 1 (1934) 63-89 [doi:10.2307/1989708, jstor:1989708, pdf]

• Hassler Whitney: Differentiable Functions Defined in Closed Sets. I, Transactions of the AMS 36 2 (1934) 369-387 [doi:10.2307/1989844, jstor:1989844]

Exposition:

• Benjamin Barr: Classical Whitney extension theorem and other preliminarites, section 1 of: On Three Theorems for Extensions of Functions, PhD thesis, Cincinatti (2023) [pdf, pdf]

Textbook accounts:

• Elias M. Stein: Extension Theorems of Whitney Type, section VI.2 of: Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, Princeton University Press (1971) [ISBN:9780691080796]

• Lars Hörmander, Theorem 2.3.6 of: The analysis of linear partial differential operators, vol. I, Springer (1983, 1990) [pdf]

  (only treats extension from compact subsets)

• Lawrence C. Evans, Ronald F. Gariepy: Whitney’s Extension Theorem, section 6.5 in: Measure Theory and Fine Properties of Functions, Textbooks in Mathematics, CRC Press (2015) [ISBN:9781032946443, pdf]

• Boris M. Makarov , Anatolii N. Podkorytov, Chapter 11 of: Smooth Functions and Maps, Moscow Lectures 7, Springer (2021) [doi:10.1007/978-3-030-79438-5]

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]

Further discussion:

• Charles Fefferman: A sharp form of Whitney’s extension theorem, Ann. Math. 161 1 (2005) 509-577 [doi:10.4007/annals.2005.161.509]

• Charles Fefferman: Whitney’s Extension Problems and Interpolation of Data, Bull. Amer. Math. Soc. 46 (2009), 207-220 [doi:10.1090/S0273-0979-08-01240-8, pdf]

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

• David Michael Roberts, Alexander Schmeding: Extending Whitney’s extension theorem: nonlinear function spaces, Annales de l’Institut Fourier 71 3 (2021) 1241-1286 [arXiv:1801.04126, numdam:10.5802/aif.3424]

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).