TY - JOUR

T1 - Computability in Harmonic Analysis

AU - Binder, Ilia

AU - Glucksam, Adi

AU - Rojas, Cristobal

AU - Yampolsky, Michael

N1 - Funding Information:
I.B. and M.Y. were partially supported by NSERC Discovery grants. A.G. was partially supported by Schmidt Futures program. C.R was partially supported by the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska-Curie Grant Agreement No. 731143 and by ANID under the Fondecyt Regular Project No. 1190493.
Publisher Copyright:
© 2021, SFoCM.

PY - 2022/6

Y1 - 2022/6

N2 - 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.

AB - 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.

KW - Computable analysis

KW - Harmonic measure

KW - Piece-wise computable non-computable functions

UR - http://www.scopus.com/inward/record.url?scp=85110372730&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85110372730&partnerID=8YFLogxK

U2 - 10.1007/s10208-021-09524-w

DO - 10.1007/s10208-021-09524-w

M3 - Article

AN - SCOPUS:85110372730

SN - 1615-3375

VL - 22

SP - 849

EP - 873

JO - Foundations of Computational Mathematics

JF - Foundations of Computational Mathematics

IS - 3

ER -