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Extreme harmonic functions on a ball. (English) Zbl 1048.31003
Let \(H\) denote the (convex) set of all positive harmonic functions on the unit ball \(B\) in \(\mathbb R^n\) that are valued 1 at the origin. The Riesz-Herglotz theorem gives a representation for members of \(H\) as Poisson integrals of probability measures on \(\partial B\). This result can be obtained as an application of the Krein-Milman theorem once one has established that the extreme points of \(H\) are precisely the Poisson kernels. R. R. Phelps [Lectures on Choquet’s theorem (Van Nostrand, Princeton) (1966; Zbl 0135.36203)] therefore asked for a simple verification of this fact without recourse to the Riesz-Herglotz theorem. In response the authors provide a particularly elementary proof in this attractive expository article. An alternative approach had earlier been given by D. H. Armitage [Potential Theory – ICPT 94 (de Gruyter, Berlin), 229–232 (1996; Zbl 0856.31003)].
MSC:
31B10 Integral representations, integral operators, integral equations methods in higher dimensions
46B22 Radon-Nikodým, Kreĭn-Milman and related properties
46A55 Convex sets in topological linear spaces; Choquet theory
31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
31B15 Potentials and capacities, extremal length and related notions in higher dimensions
47B99 Special classes of linear operators
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