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arXiv · 1212.4394

Padé Approximants, density of rational functions in $\bbb{A^\infty(\OO)}$ and smoothness of the integration operator

Vassili Nestoridis·Ilias Zadik
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Abstract

First we establish some generic universalities for Padé approximants in the closure $X^\infty(\OO)$ in $A^\infty(\OO)$ of all rational functions with poles off $\oO$, the closure taken in $\C$ of the domain $\OO\subset\C$.\ Next we give sufficient conditions on $\OO$ so that $X^\infty(\OO)=A^\infty(\OO)$.\ Some of these conditions imply that, even if the boundary $\partial\OO$ of a Jordan domain $\OO$ has infinite length, the integration operator on $\OO$ preserves $H^\infty(\OO)$ and $A(\OO)$ as well.\ We also give an example of a Jordan domain $\OO$ and a function $f\in A(\OO)$, such that its antiderivative is not bounded on $\OO$.\ Finally we restate these results for Volterra operators on the open unit disc $D$ and we complete them by some generic results.

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Vassili Nestoridis, Ilias Zadik. 2012-12-18. Padé Approximants, density of rational functions in $\bbb{A^\infty(\OO)}$ and smoothness of the integration operator. https://arxiv.org/abs/1212.4394

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