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Sets of Minimal Capacity and Extremal Domains

TL;DR: 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.
Citations
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Journal ArticleDOI
TL;DR: In this paper, it was shown that the limit zero distribution of the Heine-Stieltjes polynomials is a continuous critical measure, and the notion of continuous critical measures was introduced.
Abstract: We investigate the asymptotic zero distribution of Heine-Stieltjes polynomials – polynomial solutions of second order differential equations with complex polynomial coefficients. In the case when all zeros of the leading coefficients are all real, zeros of the Heine-Stieltjes polynomials were interpreted by Stieltjes as discrete distributions minimizing an energy functional. In a general complex situation one deals instead with a critical point of the energy. We introduce the notion of discrete and continuous critical measures (saddle points of the weighted logarithmic energy on the plane), and prove that a weak-* limit of a sequence of discrete critical measures is a continuous critical measure. Thus, the limit zero distributions of the Heine-Stieltjes polynomials are given by continuous critical measures. We give a detailed description of such measures, showing their connections with quadratic differentials. In doing that, we obtain some results on the global structure of rational quadratic differentials on the Riemann sphere that have an independent interest.

112 citations

Book ChapterDOI
01 Jan 1984
TL;DR: The invariant line element was introduced in Section 5.3 and the local properties of the corresponding metric were investigated in Sections 5.4 and 8.1 as discussed by the authors, and its global properties were studied in Section 6.
Abstract: The invariant line element |φ(z)|1/2|dz| was introduced in Section 5.3 and the local properties of the corresponding metric were investigated in Sections 5.4 and 8. In this chapter, we study its global properties.

87 citations

Posted Content
TL;DR: In this article, the authors studied the existence problem of a class of systems of curves in the complex plane whose equilibrium potential in a harmonic external field satisfies a special symmetry property (S$-property) and proved a version of existence theorem for the case when both the set of singularities of the external field and the sets of fixed points of the class of curves are small.
Abstract: This paper is devoted to a study of $S$-curves, that is systems of curves in the complex plane whose equilibrium potential in a harmonic external field satisfies a special symmetry property ($S$-property). Such curves have many applications. In particular, they play a fundamental role in the theory of complex (non-hermitian) orthogonal polynomials. One of the main theorems on zero distribution of such polynomials asserts that the limit zero distribution is presented by an equilibrium measure of an $S$-curve associated with the problem if such a curve exists. These curves are also the starting point of the matrix Riemann-Hilbert approach to srtong asymptotics. Other approaches to the problem of strong asymptotics (differential equations, Riemann surfaces) are also related to $S$-curves or may be interpreted this way. Existence problem $S$-curve in a given class of curves in presence of a nontrivial external field presents certain challenge. We formulate and prove a version of existence theorem for the case when both the set of singularities of the external field and the set of fixed points of a class of curves are small (in main case -- finite). We also discuss various applications and connections of the theorem.

64 citations


Cites background from "Sets of Minimal Capacity and Extrem..."

  • ...The main result in [51] is the existence and uniqueness theorem of a compact S ⊂ Tf with (2....

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  • ...In his recent paper [51] he introduced a more general concept of “extremal cuts” (or maximal domain) related to an element f of an analytic function at ∞....

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Journal ArticleDOI
TL;DR: In this paper, the authors formalized the idea of singularity-based extrapolation and showed that significant improvements can be achieved using exactly the same input data, and illustrate the general method with examples from quantum mechanics and quantum field theory.

58 citations

References
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Book
01 Jan 1929

2,246 citations


"Sets of Minimal Capacity and Extrem..." refers background in this paper

  • ...holds true (see Chapter VIII of [11] or [8], Theorem 1....

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Book
01 Jan 1972
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.
Abstract: 1. Spaces of measures and signed measures. Operations on measures and signed measures (No. 1-5).- 2. Space of distributions. Operations on distributions (No. 6-10)..- 3. The Fourier transform of distributions (No. 11-13).- I. Potentials and their basic properties.- 1. M. Riesz kernels (No. 1-3).- 2. Superharmonic functions (No. 4-5).- 3. Definition of potentials and their simplest properties (No. 6-9)...- 4. Energy. Potentials with finite energy (No. 10-15).- 5. Representation of superharmonic functions by potentials (No. 16-18).- 6. Superharmonic functions of fractional order (No. 19-25).- II. Capacity and equilibrium measure.- 1. Equilibrium measure and capacity of a compact set (No. 1-5).- 2. Inner and outer capacities and equilibrium measures. Capacitability (No. 6-10).- 3. Metric properties of capacity (No. 11-14).- 4. Logarithmic capacity (No. 15-18).- III. Sets of capacity zero. Sequences and bounds for potentials.- 1. Polar sets (No. 1-2).- 2. Continuity properties of potentials (No. 3-4).- 3. Sequences of potentials of measures (No. 5-8).- 4. Metric criteria for sets of capacity zero and bounds for potentials (No. 9-11).- IV. Balayage, Green functions, and the Dirichlet problem.- 1. Classical balayage out of a region (No. 1-6).- 2. Balayage for arbitrary compact sets (No. 7-11).- 3. The generalized Dirichlet problem (No. 12-14).- 4. The operator approach to the Dirichlet problem and the balayage problem (No. 15-18).- 5. Balayage for M. Riesz kernels (No. 19-23)...- 6. Balayage onto Borel sets (No. 24-25).- V. Irregular points.- 1. Irregular points of Borel sets. Criteria for irregularity (No. 1-6)...- 2. The characteristics and types of irregular points (No. 7-8)...- 3. The fine topology (No. 9-11).- 4. Properties of set of irregular points (No. 12-15).- 5. Stability of the Dirichlet problem. Approximation of continuous functions by harmonic functions (No. 16-22).- VI. Generalizations.- 1. Distributions with finite energy and their potentials (No. 1-5)...- 2. Kernels of more general type (No. 6-11).- 3. Dirichlet spaces (No. 12-15).- Comments and bibliographic references.

1,885 citations


"Sets of Minimal Capacity and Extrem..." refers background or methods or result in this paper

  • ...The lemma can be proved like the analogous result in [15], Theorem 1....

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  • ...Part (ii) of the lemma has been proved in [15], Theorem 1....

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  • ...There exist techniques to overcome this specific problem, as for instance, the use of local smoothing techniques at the singularity of the Green function, which is demonstrated in detail in [15], Chapter 1, §5....

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  • ...As general references to potentialtheoretic results we have used [27], [26], and sometimes also [15]....

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Book
27 Aug 1992
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.
Abstract: 1. Some Basic Facts.- 2. Continuity and Prime Ends.- 3. Smoothness and Corners.- 4. Distortion.- 5. Quasidisks.- 6. Linear Measure.- 7. Smirnov and Lavrentiev Domains.- 8. Integral Means.- 9. Curve Families and Capacity.- 10. Hausdorff Measure.- 11. Local Boundary Behaviour.- References.- Author Index.

1,863 citations

Book
01 Jan 1969

1,598 citations


Additional excerpts

  • ...We also mention in this respect the textbooks [5] and [24]....

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Book
01 Jan 1997
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.
Abstract: This treatment of potential theory emphasizes the effects of an external field (or weight) on the minimum energy problem. Several important aspects of the external field problem (and its extension to signed measures) justify its special attention. The most striking is that it provides a unified approach to seemingly different problems in constructive analysis. These include the asymptotic analysis of orthogonal polynomials, the limited behavior of weighted Fekete points; the existence and construction of fast decreasing polynomials; 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.

1,560 citations


"Sets of Minimal Capacity and Extrem..." refers background or methods or result in this paper

  • ...As general references to potentialtheoretic results we have used [27], [26], and sometimes also [15]....

    [...]

  • ...5 in Chapter II of [27] it follows that if we define the measure σ on γ by dσ(v) := − 1 2π ( ∂ ∂n− + ∂ ∂n+ )u(v)dsv, (11....

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  • ...A systematic investigation of Green energy and Green capacity together with the associated Green potentials can be found in [27], Chapter II....

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  • ...1 in the Appendix A of [27]) that d(z) = 0 for z ∈ D, which proves the lemma....

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  • ...1 of [27], which will also be used in the present subsection; it is the basis for the proof of Lemma 36, below, after the next paragraph....

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