|
|
| Line 1: |
Line 1: |
| 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.'''
| | The title of the writer is Figures. Her family members lives in Minnesota. Since she was eighteen she's been operating as a receptionist but her marketing never arrives. It's not a typical thing but what she likes doing is base jumping and now she is attempting to earn cash with it.<br><br>Feel free to visit my blog :: home std test kit - [http://jewelrycase.co.kr/xe/Ring/11593 have a peek at this website], |
| | |
| == 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''<sup>m</sup> function ''f''('''x''') on '''R'''<sup>n</sup>, Taylor's theorem asserts that for each '''a''', '''x''', '''y''' ∈ '''R'''<sup>n</sup>, it is possible to write
| |
| {{NumBlk|::|<math>f({\bold x}) = \sum_{|\alpha|\le m} \frac{D^\alpha f({\bold y})}{\alpha!}\cdot ({\bold x}-{\bold y})^{\alpha}+\sum_{|\alpha|=m} R_\alpha({\bold x},{\bold y})\frac{({\bold x}-{\bold y})^\alpha}{\alpha!}</math>|{{EquationRef|1}}}}
| |
| where α is a [[multi-index]] and ''R''<sub>α</sub>('''x''','''y''') → 0 uniformly as '''x''','''y''' → '''a'''.
| |
| | |
| Let ''f''<sub>α</sub>=''D''<sup>α</sup>''f'' for each multi-index α. Differentiating (1) with respect to '''x''', and possibly replacing ''R'' as needed, yields
| |
| {{NumBlk|::|<math>f_\alpha({\bold x})=\sum_{|\beta|\le m-|\alpha|}\frac{f_{\alpha+\beta}({\bold y})}{\beta!}({\bold x}-{\bold y})^{\beta}+R_\alpha({\bold x},{\bold y})</math>|{{EquationRef|2}}}}
| |
| where ''R''<sub>α</sub> is ''o''(|'''x'''-'''y'''|<sup>m-|α|</sup>) uniformly as '''x''','''y''' → '''a'''.
| |
| | |
| Note that ({{EquationNote|2}}) may be regarded as purely a compatibility condition between the functions ''f''<sub>α</sub> 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''<sub>α</sub> are a collection of functions on a closed subset ''A'' of '''R'''<sup>n</sup> for all multi-indices α with <math>|\alpha|\le m</math> satisfying the compatibility condition ({{EquationNote|2}}) at all points ''x'', ''y'', and ''a'' of ''A''. Then there exists a function ''F''('''x''') of class ''C''<sup>m</sup> such that: | |
| # ''F''=''f''<sub>0</sub> on ''A''.
| |
| # ''D''<sup>α</sup>''F'' = ''f''<sub>α</sub> on ''A''.
| |
| # ''F'' is real-analytic at every point of '''R'''<sup>n</sup>-''A''.
| |
| | |
| Proofs are given in the original paper of {{harvtxt|Whitney|1934}}, as well as in {{harvtxt|Malgrange|1967}}, {{harvtxt|Bierstone|1980}} and {{harvtxt|Hörmander|1990}}.
| |
| | |
| ==Extension in a half space==
| |
| {{harvtxt|Seeley|1964}} proved a sharpening of the Whitney extension theorem in the special case of a half space. A smooth function on a half space '''R'''<sup>''n'',+</sup> of points where ''x''<sub>''n''</sub> ≥ 0 is a smooth function ''f'' on the interior ''x''<sub>''n''</sub> for which the derivatives ∂<suP>α</sup> ''f'' extend to continuous functions on the half space. On the boundary ''x''<sub>''n''</sub> = 0, ''f'' restricts to smooth function. By [[Borel's lemma]] can be extended to a
| |
| smooth function on the whole of '''R'''<sup>''n''</sub>. Since Borel's lemms is local in nature, the same argument shows that if Ω is a (bounded or unbounded) domain in '''R'''<sup>''n''</sub> with smooth boundary, then any smooth function on the closure of Ω can be extended to a smooth function on '''R'''<sup>''n''</sup>.
| |
| | |
| Seeley's result for a half line gives a uniform extension map
| |
| | |
| :<math>\displaystyle{E:C^\infty(\mathbf{R}^+)\rightarrow C^\infty(\mathbf{R}),}</math>
| |
| | |
| which is linear, continuous (for the topology of uniform convergence of functions and their derivatives on compacta) and takes functions supported in [0,''R''] into functions supported in [−''R'',''R'']
| |
| | |
| To define ''E'', set<ref>{{harvnb|Bierstone|1980|p=143}}</ref>
| |
| | |
| :<math>\displaystyle{E(f)(x)=\sum_{m=1}^\infty a_m f(-b_mx)\varphi(-b_mx) \,\,\, (x < 0),}</math>
| |
| | |
| where φ is a smooth function of compact support on ''R'' equal to 1 near 0 and the sequences (''a''<sub>''m''</sub>), (''b''<sub>''m''</sub>) satisfy:
| |
| | |
| *''b''<sub>''m''</sub> > 0 tends to ∞;
| |
| *∑ ''a''<sub>''m''</sub> ''b''<sub>''m''</sub><sup>''j''</sup> = (−1)<sup>''j''</sup> for ''j'' ≥ 0 with the sum absolutely convergent.
| |
| | |
| A solution to this system of equations can be obtained by taking ''b''<sub>''n''</sub> = 2<sup>''n''</sup> and seeking an [[entire function]]
| |
| | |
| :<math>\displaystyle{g(z)=\sum_{m=1}^\infty a_m z^m}</math> | |
| | |
| such that ''g''(2<sup>''j''</sup>) = (−1)<sup>''j''</sup>. That such a function can be constructed follows from the [[Weierstrass theorem]] and [[Mittag-Leffler theorem]].<ref>{{harvnb|Ponnusamy|Silverman|2006|pp=442–443}}</ref>
| |
| | |
| It can be seen directly by setting<ref>{{harvnb|Chazarain|Piriou|1982}}</ref>
| |
| | |
| :<math>\displaystyle{W(z)=\prod_{j\ge 1} (1-z/2^j),}</math>
| |
| | |
| an entire function with simple zeros at 2<sup>''j''</sup>. The derivatives ''W'' '(2<sup>''j''</sup>) are bounded above and below. Similarly the function
| |
| | |
| :<math>\displaystyle{M(z)=\sum_{j\ge 1} {(-1)^j\over W^\prime(2^j) (z-2^j)}}</math> | |
| | |
| meromorphic with simple poles and prescribed residues at 2<sup>''j''</sup>.
| |
| | |
| By construction
| |
| | |
| :<math>\displaystyle{g(z)=W(z)M(z)}</math>
| |
| | |
| is an entire function with the required properties.
| |
| | |
| The definition for a half space in '''R'''<sup>''n''</sup> by applying the operator ''R'' to the last variable ''x''<sub>''n''</sub>. Similarly, using a smooth [[partition of unity]] and a local change of variables, the result for a half space implies the existence of an analogous extending map
| |
| | |
| :<math>\displaystyle{C^\infty(\overline{\Omega}) \rightarrow C^\infty(\mathbf{R}^n)}</math>
| |
| | |
| for any domain Ω in '''R'''<sup>''n''</sup> with smooth boundary.
| |
| | |
| ==Notes==
| |
| {{reflist|2}}
| |
| | |
| ==References==
| |
| * Extension of range of functions, Edward James McShane, Bull. Amer. Math. Soc., 40:837-842, 1934. {{MR|1562984}}
| |
| * {{citation|title=Analytic extensions of functions defined in closed sets|first=Hassler|last=Whitney|authorlink=Hassler Whitney|journal=Transactions of the American Mathematical Society|year=1934|volume=36|pages=63–89|doi=10.2307/1989708|jstor=1989708|issue=1|publisher=American Mathematical Society}}
| |
| *{{citation|journal=Bulletin of the Brazilian Mathematical Society|volume=11|year=1980|pages= 139–189|
| |
| title=Differentiable functions|first=Edward|last= Bierstone|authorlink=Edward Bierstone|url=http://www.springerlink.com/content/v6h426156663k441/}}
| |
| *{{citation|last=Malgrange|first= Bernard|title= Ideals of differentiable functions|series= Tata Institute of Fundamental Research Studies in Mathematics|volume=3| publisher=Oxford University Press|year=1967}}
| |
| *{{citation|last=Seeley|first= R. T.|title= Extension of C∞ functions defined in a half space|journal=Proc. Amer. Math. Soc. |volume=15|year= 1964 |pages=625–626}}
| |
| *{{citation|last=Hörmander|first= Lars|title= The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis|publisher= Springer-Verlag|year=1990|isbn=3-540-00662-1}}
| |
| *{{citation|title=Introduction to the Theory of Linear Partial Differential Equations|volume=14|series= Studies in Mathematics and Its Applications|first=Jacques|last= Chazarain|first2= Alain|last2= Piriou|publisher=Elsevier|year= 1982|isbn=0444864520}}
| |
| *{{citation|last=Ponnusamy|first= S.|last2= Silverman|first2= Herb|title= Complex variables with applications|publisher=Birkhäuser|year=2006|isbn= 0-8176-4457-1}}
| |
| | |
| [[Category:Theorems in analysis]]
| |
The title of the writer is Figures. Her family members lives in Minnesota. Since she was eighteen she's been operating as a receptionist but her marketing never arrives. It's not a typical thing but what she likes doing is base jumping and now she is attempting to earn cash with it.
Feel free to visit my blog :: home std test kit - have a peek at this website,