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

Burgess, Walter D. ; Raphael, Robert M.

Czechoslovak Mathematical Journal, Tome 73 (2023) no. 2, pp. 487-498.

Résumé

For $k\in {\mathbb N} \cup \{\infty \}$ 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 ${\rm Q}_{\rm cl} ({\rm C}^{k} ({\mathbb R})) = {\rm Q}({\rm C}^{k} ({\mathbb R}))$.

Détail

DOI : 10.21136/CMJ.2023.0445-21

Classification : 13B30, 26A24, 54C30
Keywords: ${\rm C}^k$ function; spline; ring of quotient; Mollifier function