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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 related result is due to McShane, hence it is sometimes called the McShane–Whitney extension theorem. == Statement == 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 R''n'', Taylor's theorem asserts that for each a, x, y ∈ R''n'', there is a function ''R''''α''(x,y) approaching 0 uniformly as x,y → a such that where the sum is over multi-indices ''α''. Let ''f''''α'' = ''D''''α''''f'' for each multi-index ''α''. Differentiating (1) with respect to x, and possibly replacing ''R'' as needed, yields where ''R''''α'' is ''o''(|x − y|''m''−|''α''|) uniformly as x,y → a. Note that () may be regarded as purely a compatibility condition between the functions ''f''α which must be satisfied in order for these functions to be the coefficients of the Taylor series of the function ''f''. It is this insight which facilitates the following statement Theorem. Suppose that ''f''''α'' are a collection of functions on a closed subset ''A'' of R''n'' for all multi-indices α with satisfying the compatibility condition () at all points ''x'', ''y'', and ''a'' of ''A''. Then there exists a function ''F''(x) of class ''C''''m'' such that: # ''F'' = ''f''0 on ''A''. # ''D''''α''''F'' = ''f''''α'' on ''A''. # ''F'' is real-analytic at every point of R''n'' − ''A''. Proofs are given in the original paper of , and in , and . 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Whitney extension theorem」の詳細全文を読む スポンサード リンク
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