Motivated by the classical Dirichlet problem, the authors prove that for every open set \(U\) in an arbitrary Riemann surface and every continuous function \(\varphi\) on the boundary \(\partial U\), there exists a holomorphic function \(\widetilde{\varphi}\) on \(U\) such that \(\forall p\in\partial U, \ \widetilde{\varphi}(x)\xrightarrow{} \varphi(p)\), as \(x \to p\) outside a set of density 0 at \(p \) relative to \(U\).
First they slighty extend a result of A. Boivin [Math. Ann. 275, 57–70 (1986; Zbl 0583.30035)] which characterizes the sets of tangential approximation in an open Riemann surface and then they prove that every chaplet \(E\) of an open Riemann surface \(M\) is a set of tangential approximation. The latter result plays a central role to obtain their goal.
Finally, they also present the following interesting approximation result, where the approximation function has interpolation and Picard-type properties: Let \(M\) be an open Riemann surface, let \(E\) be a chaplet in an open subset \(U\subset M\) and let \((x_n)_n\) be a sequence of distinct points in \( U \smallsetminus E\) with no accumulation points in \(U\). Then for every function \( f\in A(E)\), every sequence \((y_n)_n\subset\mathbb{C}\) and every positive and continuous function \(\varepsilon\) on \(E\), there exists a function \(g\) holomorphic in \(U\) such that \( g(x_n) = y_n, \ \forall n\in\mathbb{N}\) and \(|g(x)-f(x)| <\varepsilon (x), \ \forall x\in E\). Furthermore, g can be chosen so that for every \(p\in\partial U\) and for every complex number \(\beta\) there exists a sequence \((a_{j})_j\) in \(U\smallsetminus E\) with \(a_j\to p\), when \(j\) goes to infinity and \(g(a_j) = \beta\) for each natural number \(j\).