Computability in Harmonic Analysis

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.