Computability in Harmonic Analysis

Ilia Binder, Adi Glucksam, Cristobal Rojas*, Michael Yampolsky

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

We study the question of constructive approximation of the harmonic measure ωxΩ of a bounded domain Ω with respect to a point x∈ Ω. In particular, using a new notion of computable harmonic approximation, we show that for an arbitrary such Ω, computability of the harmonic measure ωxΩ for a single point x∈ Ω implies computability of ωyΩ for any y∈ Ω. 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.

Original languageEnglish (US)
Pages (from-to)849-873
Number of pages25
JournalFoundations of Computational Mathematics
Volume22
Issue number3
DOIs
StatePublished - Jun 2022
Externally publishedYes

Keywords

  • Computable analysis
  • Harmonic measure
  • Piece-wise computable non-computable functions

ASJC Scopus subject areas

  • Analysis
  • Computational Mathematics
  • Computational Theory and Mathematics
  • Applied Mathematics

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