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Local behaviour of the error in the Bergman kernel method for numerical conformal mapping
Minimal Riesz Energy Point Configurations for Rectifiable d-Dimensional Manifolds
For a compact set A in Euclidean space we consider the asymptotic behavior of
optimal (and near optimal) N-point configurations that minimize the Riesz
s-energy (corresponding to the potential 1/t^s) over all N-point subsets of A,
where s>0. For a large class of manifolds A having finite, positive
d-dimensional Hausdorff measure, we show that such minimizing configurations
have asymptotic limit distribution (as N tends to infinity with s fixed) equal
to d-dimensional Hausdorff measure whenever s>d or s=d. In the latter case we
obtain an explicit formula for the dominant term in the minimum energy. Our
results are new even for the case of the d-dimensional sphere.Comment: paper: 29 pages and addendum: 4 page
Geometric overconvergence of rational functions in unbounded domains
The basic aim of this paper is to study the phenomenon of overconvergence for rational functions converging geometrically on [0, + ∞)
Reverse Triangle Inequalities for Riesz Potentials and Connections with Polarization
We study reverse triangle inequalities for Riesz potentials and their
connection with polarization. This work generalizes inequalities for sup norms
of products of polynomials, and reverse triangle inequalities for logarithmic
potentials. The main tool used in the proofs is the representation for a power
of the farthest distance function as a Riesz potential of a unit Borel measure
Asymptotic behaviour of zeros of bieberbach polynomials
AbstractLet Ω be a simply-connected domain in the complex plane and let πn denote the nth-degree Bieberbach polynomial approximation to the conformal map f of Ω onto a disc. In this paper we investigate the asymptotic behaviour (as n→σ) of the zeros of πn, πn′ and also of the zeroes of certain closely related rational approximants to f. Our result show that, in each case, the distribution of the zeros is governed by the location of the singularities of the mapping function f in C⧹ω, and we present numerical examples illustrating this
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