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Sets of Minimal Capacity and Extremal Domains
TLDR
In this article, a unique existence theorem for extremal domains and their complementary sets of minimal capacity is proved, and analytic tools for their characterization are presented; most notable are here quadratic differentials and a specific symmetry property of the Green function in the extremal domain.Abstract:
Let f be a function meromorphic in a neighborhood of infinity. The central problem in the present investigation is to find the largest domain D \subset C to which the function f can be extended in a meromorphic and singlevalued manner. 'Large' means here that the complement C\D is minimal with respect to (logarithmic) capacity. Such extremal domains play an important role in Pad'e approximation. In the paper a unique existence theorem for extremal domains and their complementary sets of minimal capacity is proved. The topological structure of sets of minimal capacity is studied, and analytic tools for their characterization are presented; most notable are here quadratic differentials and a specific symmetry property of the Green function in the extremal domain. A local condition for the minimality of the capacity is formulated and studied. Geometric estimates for sets of minimal capacity are given. Basic ideas are illustrated by several concrete examples, which are also used in a discussion of the principal differences between the extremality problem under investigation and some classical problems from geometric function theory that possess many similarities, which for instance is the case for Chebotarev's Problem.read more
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References
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Book
Foundations of Modern Potential Theory
TL;DR: In this paper, the authors define the notion of potentials and their basic properties, including the capacity and capacity of a compact set, the properties of a set of irregular points, and the stability of the Dirichlet problem.
Book
Boundary Behaviour of Conformal Maps
TL;DR: In this paper, the authors describe local boundary behavior in terms of curve families, curve families and capacity, and the Hausdorff measure, which is a measure of the curve families' capacity.
Book
Logarithmic Potentials with External Fields
Edward B. Saff,Vilmos Totik +1 more
TL;DR: In this paper, the authors consider the effects of an external field (or weight) on the minimum energy problem and provide a unified approach to seemingly different problems in constructive analysis, such as the asymptotic analysis of orthogonal polynomials, the limited behavior of weighted Fekete points, the existence and construction of fast decreasing polynomial, the numerical conformal mapping of simply and doubly connected domains, generalization of the Weierstrass approximation theorem to varying weights, and the determination of convergence rates for best approximating rational functions.