McShane–Whitney extension theorem

In mathematics, in particular in mathematical analysis, the Whitney extension theorem is a partial converse to Taylor's theorem. Roughly speaking, the theorem asserts that if A is a closed subset of a Euclidean space, then it is possible to extend a given function of A in such a way as to have prescribed derivatives at the points of A. It is a result of Hassler Whitney.

A precise statement of the theorem requires careful consideration of what it means to prescribe the derivative of a function on a closed set. One difficulty, for instance, is that closed subsets of Euclidean space in general lack a differentiable structure. The starting point, then, is an examination of the statement of Taylor's theorem.

Given a real-valued C^m function f(x) on Rⁿ, Taylor's theorem asserts that for each a, x, y ∈ Rⁿ, there is a function R_α(x,y) approaching 0 uniformly as x,y → a such that

${\displaystyle f({\mathbf {x}})=\sum _{|\alpha |\leq m}{\frac {D^{\alpha }f({\mathbf {y}})}{\alpha !}}\cdot ({\mathbf {x}}-{\mathbf {y}})^{\alpha }+\sum _{|\alpha |=m}R_{\alpha }({\mathbf {x}},{\mathbf {y}}){\frac {({\mathbf {x}}-{\mathbf {y}})^{\alpha }}{\alpha !}}}$

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