General form of Whitney's Extension Theorem
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The Whitney Extension Theorem, as is stated in most sources, deals with extending functions $ f : mathbb{R}^n rightarrow mathbb{R} $ defined on closed sets, in a $ C^r $ way to the ambient space. Is there a version for functions $ f : mathbb{R}^n rightarrow mathbb{R}^m $?
Each of the component functions is a function $ mathbb{R}^n rightarrow mathbb{R} $, and the extension theorem can be applied individually. But I'm not sure this is the right notion, and I haven't found it in any of the texts that state Whitney's theorem.
real-analysis analysis
asked Jan 8 at 23:30
DanielDaniel
764
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$begingroup$
The Whitney Extension Theorem, as is stated in most sources, deals with extending functions $ f : mathbb{R}^n rightarrow mathbb{R} $ defined on closed sets, in a $ C^r $ way to the ambient space. Is there a version for functions $ f : mathbb{R}^n rightarrow mathbb{R}^m $?
Each of the component functions is a function $ mathbb{R}^n rightarrow mathbb{R} $, and the extension theorem can be applied individually. But I'm not sure this is the right notion, and I haven't found it in any of the texts that state Whitney's theorem.
real-analysis analysis
asked Jan 8 at 23:30
DanielDaniel
764
$endgroup$
add a comment |
$begingroup$
The Whitney Extension Theorem, as is stated in most sources, deals with extending functions $ f : mathbb{R}^n rightarrow mathbb{R} $ defined on closed sets, in a $ C^r $ way to the ambient space. Is there a version for functions $ f : mathbb{R}^n rightarrow mathbb{R}^m $?
Each of the component functions is a function $ mathbb{R}^n rightarrow mathbb{R} $, and the extension theorem can be applied individually. But I'm not sure this is the right notion, and I haven't found it in any of the texts that state Whitney's theorem.
real-analysis analysis
asked Jan 8 at 23:30
DanielDaniel
764
$endgroup$
The Whitney Extension Theorem, as is stated in most sources, deals with extending functions $ f : mathbb{R}^n rightarrow mathbb{R} $ defined on closed sets, in a $ C^r $ way to the ambient space. Is there a version for functions $ f : mathbb{R}^n rightarrow mathbb{R}^m $?
Each of the component functions is a function $ mathbb{R}^n rightarrow mathbb{R} $, and the extension theorem can be applied individually. But I'm not sure this is the right notion, and I haven't found it in any of the texts that state Whitney's theorem.
real-analysis analysis
real-analysis analysis
asked Jan 8 at 23:30
DanielDaniel
764
asked Jan 8 at 23:30
DanielDaniel
764
asked Jan 8 at 23:30
DanielDaniel
764
asked Jan 8 at 23:30
DanielDaniel
764
asked Jan 8 at 23:30
DanielDaniel
764
764
add a comment |
add a comment |
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