Computation of Analytic Capacity and Applications to the Subadditivity Problem
Malik Younsi,Thomas Ransford +1 more
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TLDR
In this article, a least square method for computing the analytic capacity of compact plane sets with piecewise-analytic boundary is presented. But the method is restricted to the case where the plane sets have a piecewise analytic boundary, and it is not shown that analytic capacity is subadditive.Abstract:
We develop a least-squares method for computing the analytic capacity of compact plane sets with piecewise-analytic boundary. The method furnishes rigorous upper and lower bounds which converge to the true value of the capacity. Several illustrative examples are presented. We are led to formulate a conjecture which, if true, would imply that analytic capacity is subadditive. The conjecture is proved in a special case.read more
Citations
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Journal ArticleDOI
Real and Complex Analysis. By W. Rudin. Pp. 412. 84s. 1966. (McGraw-Hill, New York.)
TL;DR: In this paper, the Riesz representation theorem is used to describe the regularity properties of Borel measures and their relation to the Radon-Nikodym theorem of continuous functions.
Journal ArticleDOI
Rational Ahlfors Functions
TL;DR: In this paper, it was shown that rational Ahlfors functions of degree two are characterized by having positive residues at their poles and that this characterization does not generalize to higher degrees, with the help of a numerical method for the computation of analytic capacity.
Journal ArticleDOI
Continuity of capacity of a holomorphic motion
TL;DR: In this paper, the behavior of various set-functions under holomorphic motions is studied and it is shown that under such deformations, logarithmic capacity varies continuously, while analytic capacity may not.
Journal ArticleDOI
On the analytic and Cauchy capacities
TL;DR: In this paper, the Ahlfors function is not the Cauchy transform of any complex Borel measure supported on the set, and the authors give sufficient conditions for a compact set E ⊆ C to satisfy γ(E) = γc(E).
References
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Book
Real and complex analysis
TL;DR: In this paper, the Riesz representation theorem is used to describe the regularity properties of Borel measures and their relation to the Radon-Nikodym theorem of continuous functions.
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
A course of modern analysis; an introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions
TL;DR: The theory of Riemann integration as mentioned in this paper is a generalization of the theory of complex numbers, and it can be expressed as follows: 1. Complex numbers 2. The theory of convergence 3. Continuous functions and uniform convergence 4. The fundamental properties of analytic functions 5. The expansion of functions in infinite series 6.