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

TL;DR

This paper explores the universality of Padé approximants in certain function spaces, provides conditions for density of rational functions, examines the smoothness of the integration operator, and discusses implications for Volterra operators.

Contribution

It establishes generic universality results for Padé approximants, characterizes when rational functions are dense in specific function spaces, and analyzes the smoothness of the integration operator on Jordan domains.

Findings

Padé approximants exhibit generic universality in certain function spaces.

Conditions are identified under which rational functions are dense in $A^ abla(\\Omega)$.

The integration operator can preserve boundedness and smoothness under specific boundary conditions.

Abstract

First we establish some generic universalities for Pad\'{e} 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.