Approximation of a Function and its Derivatives by Entire Functions. Issue 1 (1st March 2016)

• Record Type: Journal Article
• Authors: Gauthier, Paul M.; Kienzle, Julie
• Is Part Of: Canadian mathematical bulletin =. Volume 59:Issue 1(2016)
• Page Start: 87
• Page End: 94
• Publication Date: 2016-03-01
• Subjects: 30E10; Carleman theorem; Mathematics -- Periodicals; Mathematics; Periodicals; 510.5
• DOI: 10.4153/CMB-2015-060-5
• Languages: English
• ISSNs: 0008-4395

Abstract

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.