A remark on the approximation theorems of Whitney and Carleman-Scheinberg

Abstract

We show that a $C^k$-smooth mapping on an open subset of $\mathbb R^n$, $k\in \mathbb N\cup\{0,\infty\}$, can be approximated in a fine topology and together with its derivatives by a restriction of a holomorphic mapping with explicitly described domain. As a corollary we obtain a generalisation of the Carleman-Scheinberg theorem on approximation by entire functions.