Computability in Harmonic Analysis

TL;DR

This paper investigates the computability of harmonic measures in bounded domains, establishing that computability at one point implies computability everywhere and exploring conditions for uniform computability.

Contribution

It introduces a new notion of computable harmonic approximation and characterizes when harmonic measures are uniformly computable for regular domains.

Findings

Computability at one point implies computability at all points in the domain.

Existence of functions computable at every point but requiring different algorithms.

Characterization of conditions for uniform computability of harmonic measure.

Abstract

We study the question of constructive approximation of the harmonic measure $\omega_x^\Omega$ of a connected bounded domain $\Omega$ with respect to a point $x\in\Omega$. 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 $x\in\Omega$ implies computability of $\omega_y^\Omega$ for any $y\in \Omega$. 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 domain. 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.