La capacitat analı́tica en problemes d’aproximació racional

Abstract

[en] This paper studies the relationship, depending on the compact set $K \subset \mathbb{C}$, between the family of continuous functions on $K, \mathcal{C}(K)$, the family of continuous functions on $K$ and analytics on $\overset{\circ}{K}, \mathcal{A}(K)$, the family of uniformly approximable functions on $K$ by rational functions with poles out on $K, \mathcal{R}(K)$, and the family of uniformly approximable functions on $K$ by polynomials, $\mathcal{P}(K)$.

We will see that it is easy to characterise $K$ in order to achive $\mathcal{P}(K)=\mathcal{R}(K)$ or $\mathcal{A}(K)=\mathcal{C}(K)$, but it is more complicated to do the same in order to achieve $\mathcal{R}(K)=\mathcal{A}(K)$.

In order to see all the possible relationships, we present some new concepts like the Hausdorff measure, content and dimension, the analytic capacity and the continuous analytic capacity.

The main part of this essay is focused on the Vitushkin Theorem, which allows us to characterise the compacts $K$, such as $\mathcal{R}(K)=\mathcal{A}(K)$. we present a demostration scheme and the results obtained from it. In addition, we will also state the Inner Boundary Conjecture that provides us with the sufficient condition on $K$ to ensure that $\mathcal{R}(K)=\mathcal{A}(K)$.