Revistes Catalanes amb Accés Obert (RACO)

Mergelyan type theorems for some function spaces

A. Stray


Let $F$ be a relatively closed subset of the unit disc $D$. If $A$ is any of the Hardy spaces $H^p(D)$, $0 < p < \infty$, $\overline{A|_F}$ denotes the functions on $F$ being uniform limits of elements from $H^p(D)$. Let $\tilde F$ consist of all $z\in D$ such that $|f(z)|\le\sup \{|f(z)| z\in F\}$ for any bounded analytic function in $D$. It is proved that $\overline{A|_F}$ consist of all functions $f$ that can be decomposed as $f=u+v$, where $u$ belongs to $H^p(D)$ and $v$ is a uniformly continuous function on the set $\tilde F$, analytic at interior points of $\tilde F$.

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