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Rational approximation on $A^{\infty }(\varOmega )$

Volume 248 / 2019

Javier Falcó, Vassili Nestoridis Studia Mathematica 248 (2019), 93-107 MSC: Primary 32A30, 30E10; Secondary 32A38. DOI: 10.4064/sm171109-5-3 Published online: 18 December 2018

Abstract

Let $\{U_i\}_{i\in I}$ be a family of bounded planar domains in $\mathbb C$. Here $A(\prod_{i\in I}U_i)$ stands for the set of continuous functions on $\prod_{i\in I}\overline{U_i}$ endowed with the product topology that are separately holomorphic on each $U_i$. We consider the algebra $A^\infty(\prod_{i\in I}U_i)$ consisting of those functions in $A(\prod_{i\in I}U_i)$ all of whose mixed partial derivatives are also in $A(\prod_{i\in I}U_i)$. We show that under certain conditions the set of functions that are finite sums of finite products of rational functions of one variable whose poles are prescribed elements of $\overline{U}_i^c$ is dense in $A^\infty(\prod_{i\in I}U_i)$ with the topology of uniform convergence for every finite set of mixed partial derivatives.

Authors

  • Javier FalcóDepartamento de Análisis Matemático
    Universidad de Valencia
    Doctor Moliner 50
    46100 Burjasot (Valencia), Spain
    e-mail
  • Vassili NestoridisDepartment of Mathematics
    University of Athens
    157 84 Panepistemiopolis
    Athens, Greece
    e-mail

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