Computability in Harmonic Analysis

Abstract

We study the question of constructive approximation of the harmonic measure omega(Omega)(x) of a bounded domain Omega with respect to a point x is an element of Omega. In particular, using a new notion of computable harmonic approximation, we show that for an arbitrary such Omega, computability of the harmonic measure omega(Omega)(x) for a single point x is an element of Omega implies computability of omega(Omega)(y) for any y is an element of Omega. 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.