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Finely Harmonic Functions need not be Quasi‐Analytic
Author(s) -
Lyons Terry
Publication year - 1984
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/16.4.413
Subject(s) - harmonic function , mathematics , neighbourhood (mathematics) , constant (computer programming) , zero (linguistics) , harmonic , open set , function (biology) , constant function , mathematical analysis , independent and identically distributed random variables , zero set , pure mathematics , quantum mechanics , physics , statistics , linguistics , philosophy , evolutionary biology , piecewise , random variable , computer science , biology , programming language
If a function f is harmonic on a connected open set ∪ ⊂ R d and is constant in a neighbourhood of one point in ∪ then it is identically constant. We give an example of a non‐constant finely harmonic function defined on E ={ z ɛC|| z |<2}\ p which is identically zero on | z |⩾1. The exceptional set P is of capacity zero so E is finely open and finely connected. This example therefore shows that finely harmonic functions are not globally determined by their local behaviour.