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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
Citations
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References
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Journal ArticleDOI
Extremal domains associated with an analytic function I
TL;DR: In this article, it was shown that there exist extremal domains D 0 with minimal condenser capacity, and that these domains are uniquely determined up to a boundary set of capacity zero.
Book ChapterDOI
The convergence of padé approximants to functions with branch points
TL;DR: In this paper, the convergence of diagonal Pade approximants to a class of functions with an even number of branch points with principal singularities of square root type was studied, and convergence in capacity was shown away from a set of arcs whose location is completely determined by the location of the branch points.
Journal ArticleDOI
Orthogonal polynomials and Padé approximants associated with a system of arcs
J Nuttall,S.R Singh +1 more
TL;DR: In this article, the convergence behavior of the diagonal sequence of the Pade table associated with a function with branch points is studied and a unique set S is constructed which consists of a number of analytic Jordan arcs ending at the branch points.