We show that if $U$ is an arbitrary open subset of a Riemann surface and $\phi$ an arbitrary continuous function on the boundary $\partial U$, then there exists a holomorphic function $\tilde{\phi }$ on $U$ such that, for every $p\in \partial U$, $\tilde{\phi }\left(x\right)\to \phi \left(p\right)$, as $x\to p$ outside a set of density 0 at $p$ relative to $U$. These “solutions to a boundary problem” are not unique. In fact they can be required to have interpolating properties and also to assume all complex values near every boundary point. Our result is new even for the unit disc.

中文翻译：

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Riemann曲面上任意开集上的渐近全纯边界问题

我们证明如果 $ü$ 是Riemann曲面的任意开放子集，并且 $\phi$ 边界上的任意连续函数 $\partial ü$，那么存在一个全纯函数 $\stackrel{〜}{\phi }$ 上 $ü$ 这样，对于每个 $p\in \partial ü$， $\stackrel{〜}{\phi }（X）\to \phi （p）$，作为 $X\to p$ 在密度为0的集合外 $p$ 关系到 $ü$。这些“边界问题的解决方案”不是唯一的。实际上，可能要求它们具有插值属性，并假设每个边界点附近的所有复数值。即使对于单元光盘，我们的结果也是新的。