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Approximation of a Function and its Derivatives by Entire Functions

Published online by Cambridge University Press:  20 November 2018

Paul M. Gauthier  and
Julie Kienzle
Paul M. Gauthier
Affiliation:
Département de mathématiques et de statistique, Université de Montréal, CP-6128 Centreville, Montréal, H3C3J7 e-mail: gauthier@dms.umontreal.ca e-mail: julie.kienzle@umontreal.ca
Julie Kienzle
Affiliation:
Département de mathématiques et de statistique, Université de Montréal, CP-6128 Centreville, Montréal, H3C3J7 e-mail: gauthier@dms.umontreal.ca e-mail: julie.kienzle@umontreal.ca
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Abstract

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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.

Keywords

30E10Carleman theorem
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 59 , Issue 1 , 01 March 2016 , pp. 87 - 94
Copyright
Copyright © Canadian Mathematical Society 2016

References

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