On extending C^k functions from an open set to R, with applications

TL;DR

This paper presents a method to extend functions of class C^k from open subsets of R to all of R using spline-based mollifiers, with implications for the structure of the ring of C^k functions.

Contribution

It introduces a novel extension technique for C^k functions employing spline mollifiers, expanding the understanding of function extension and ring properties.

Findings

Existence of extensions g in C^∞ with controlled zero set

Construction of h in C^k matching the product fg on U

Implication that the real closure of C^k equals its quotient field

Abstract

For $k\in \N\cup \{\infty\}$ and $U$ open in $ \R$, let $\C^{\,k}(U)$ be the ring of real valued functions on $U$ with the first $k$ derivatives continuous. It is shown for $f\in \C^{\,k}(U)$ there is $g\in \ck{\infty}$ with $U\sbq \coz g$ and $h\in \ck{k}$ with $fg\res_U=h\res_U$. 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 $\ck{k}$ are deduced from this, in particular that $\qcl(\ck{k}) = \text{Q}(\ck{k})$.