Approximation of a Function and its Derivatives by Entire Functions. Issue 1 (1st March 2016) - Gauthier, Paul M.; Kienzle, Julie
Abstract: A simple proof is given for the fact that for $m$ a non-negative integer, a function $f\, \in \, {C^{(m)}}\, (\mathbb{R})$, and an arbitrary positive continuous function $\in$, there is an entire function $g$ such that $\left| {{g}^{(i)}}(x)\, -\, {{f}^{(i)}}(x) \right|\, <\, \in (x)$, for all $x\, \in \, \mathbb{R}$ and for each $i\, =\, 0, \, 1\, .\, .\, .\, , \, m$ . We also consider the situation where $\mathbb{R}$ is replaced by an open interval.