Czechoslovak Mathematical Journal, Vol. 73, No. 2, pp. 487-498, 2023
On extending ${\rm C}^k$ functions from an open set to $\mathbb R$ with applications
Walter D. Burgess, Robert M. Raphael
Received December 2, 2021. Published online January 16, 2023.
Abstract: 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 ${\rm Q}_{\rm cl} ({\rm C}^k ({\mathbb R})) = {\rm Q}({\rm C}^k ({\mathbb R}))$.
Keywords: ${\rm C}^k$ function; spline; ring of quotient; Mollifier function
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Affiliations: Walter D. Burgess (corresponding author), Department of Mathematics and Statistics, University of Ottawa, Ottawa, Canada, K1N 6N5, e-mail: wburgess@uottawa.ca; Robert M. Raphael, Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Blvd. W., Montréal, Canada, H4B 1R6, e-mail: r.raphael@concordia.ca
