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Approximation of a Function and its Derivatives by Entire Functions

Published:2015-12-22
Printed: Mar 2016
• Paul M. Gauthier,
Département de mathématiques et de statistique, Université de Montréal, CP-6128 Centreville, Montréal H3C3J7
• Julie Kienzle,
Département de mathématiques et de statistique, Université de Montréal, CP-6128 Centreville, Montréal H3C3J7
 Format: LaTeX MathJax PDF

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 $\epsilon,$ there is an entire function $g,$ such that $|g^{(i)}(x)-f^{(i)}(x)|\lt \epsilon(x),$ for all $x\in\mathbb{R}$ and for each $i=0,1\dots,m.$ We also consider the situation, where $\mathbb{R}$ is replaced by an open interval.
 Keywords: Carleman theorem Carleman theorem
 MSC Classifications: 30E10 - Approximation in the complex domain

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