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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
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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 show english descriptions Approximation in the complex domain 30E10 - Approximation in the complex domain
 

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