On extending ${\mathrm{C}}^k$ functions from an open set to $ℝ$ with applications

Burgess, Walter D., and Raphael, Robert M.. "On extending ${\rm C}^{k}$ functions from an open set to $\mathbb {R}$ with applications." Czechoslovak Mathematical Journal 73.2 (2023): 487-498. <http://eudml.org/doc/299347>.

@article{Burgess2023,
abstract = {For $k\in \{\mathbb \{N\}\} \cup \lbrace \infty \rbrace $ and $U$ open in $ \{\mathbb \{R\}\}$, let $\{\rm C\}^\{k\}(U)$ be the ring of real valued functions on $U$ with the first $k$ derivatives continuous. It is shown that for $f\in \{\rm C\}^\{k\}(U)$ there is $g\in \{\rm C\}^\{\infty \} (\{\mathbb \{R\}\})$ with $U\subseteq \{\rm coz\} g$ and $h\in \{\rm C\}^\{k\} (\{\mathbb \{R\}\})$ with $fg|_U=h|_U$. The function $f$ and its $k$ derivatives are not assumed to be bounded on $U$. The function $g$ is constructed using splines based on the Mollifier function. Some consequences about the ring $\{\rm C\}^\{k\} (\{\mathbb \{R\}\})$ are deduced from this, in particular that $\{\rm Q\}_\{\rm cl\} (\{\rm C\}^\{k\} (\{\mathbb \{R\}\})) = \{\rm Q\}(\{\rm C\}^\{k\} (\{\mathbb \{R\}\}))$.},
author = {Burgess, Walter D., Raphael, Robert M.},
journal = {Czechoslovak Mathematical Journal},
keywords = {$\{\rm C\}^k$ function; spline; ring of quotient; Mollifier function},
language = {eng},
number = {2},
pages = {487-498},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On extending $\{\rm C\}^\{k\}$ functions from an open set to $\mathbb \{R\}$ with applications},
url = {http://eudml.org/doc/299347},
volume = {73},
year = {2023},
}

TY - JOUR
AU - Burgess, Walter D.
AU - Raphael, Robert M.
TI - On extending ${\rm C}^{k}$ functions from an open set to $\mathbb {R}$ with applications
JO - Czechoslovak Mathematical Journal
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 73
IS - 2
SP - 487
EP - 498
AB - For $k\in {\mathbb {N}} \cup \lbrace \infty \rbrace $ and $U$ open in $ {\mathbb {R}}$, let ${\rm C}^{k}(U)$ be the ring of real valued functions on $U$ with the first $k$ derivatives continuous. It is shown that for $f\in {\rm C}^{k}(U)$ there is $g\in {\rm C}^{\infty } ({\mathbb {R}})$ with $U\subseteq {\rm coz} g$ and $h\in {\rm C}^{k} ({\mathbb {R}})$ with $fg|_U=h|_U$. The function $f$ and its $k$ derivatives are not assumed to be bounded on $U$. The function $g$ is constructed using splines based on the Mollifier function. Some consequences about the ring ${\rm C}^{k} ({\mathbb {R}})$ are deduced from this, in particular that ${\rm Q}_{\rm cl} ({\rm C}^{k} ({\mathbb {R}})) = {\rm Q}({\rm C}^{k} ({\mathbb {R}}))$.
LA - eng
KW - ${\rm C}^k$ function; spline; ring of quotient; Mollifier function
UR - http://eudml.org/doc/299347
ER -