We study the question of constructive approximation of the harmonic measure \(\omega _x^\varOmega \) of a bounded domain \(\varOmega \) with respect to a point \(x\in \varOmega \). In particular, using a new notion of computable harmonic approximation, we show that for an arbitrary such \(\varOmega \), computability of the harmonic measure \(\omega ^\varOmega _x\) for a single point \(x\in \varOmega \) implies computability of \(\omega _y^\varOmega \) for any \(y\in \varOmega \). This may require a different algorithm for different points y, which leads us to the construction of surprising natural examples of continuous functions that arise as solutions to a Dirichlet problem, whose values can be computed at any point, but cannot be computed with the use of the same algorithm on all of their domains. We further study the conditions under which the harmonic measure is computable uniformly, that is by a single algorithm, and characterize them for regular domains with computable boundaries.

我们研究关于一个点\(x\in \varOmega \)的有界域\(\varOmega \)的调和测度\(\omega _x^\varOmega \)的建设性逼近问题。特别是，使用可计算调和近似的新概念，我们证明对于任意这样的\(\varOmega \)，单点\(x\in )的调和测度\(\omega ^\varOmega _x\) 的可计算性\varOmega \)意味着\(\omega _y^\varOmega \)对于任何\(y\in \varOmega \) 的可计算性。这可能需要针对不同的点y使用不同的算法 ，这导致我们构建了令人惊讶的连续函数的自然示例，这些示例作为狄利克雷问题的解决方案而出现，其值可以在任何点计算，但不能在所有域上使用相同的算法进行计算。我们进一步研究了谐波测度可均匀计算的条件，即通过单一算法，并针对具有可计算边界的规则域表征它们。