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Orthogonal Polynomials and $S$-curves
TLDR
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.read more
Citations
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Journal ArticleDOI
QUADRATIC DIFFERENTIALS (Ergebnisse der Mathematik und ihrer Grenzgebiete, Third Series, 5)
Journal ArticleDOI
S-curves in polynomial external fields
TL;DR: This paper gives a detailed proof of the existence of a curve with the S-property in the external field Re V within the collection of all curves that connect two or more pre-assigned directions at infinity in which Re V ?
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The distribution of the zeros of the Hermite-Padé polynomials for a pair of functions forming a Nikishin system
TL;DR: In this paper, it was shown that the zeros of the Hermite-Pade polynomials of the first kind for a pair of functions with an arbitrary even number of common branch points lying on the real axis form a generalized complex Nikishin system.
Posted Content
S-curves in Polynomial External Fields
TL;DR: In this paper, the authors give a detailed proof of the existence of a curve with the S-property in the external field given by the real part of a polynomial V, within the collection of all curves that connect two or more pre-assigned directions at infinity.
Journal ArticleDOI
Method of interior variations and existence of S -compact sets
TL;DR: In this paper, the variation of equilibrium energy is analyzed for three different functionals that naturally arise in solving a number of problems in the theory of constructive rational approximation of multivalued analytic functions.
References
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Book
Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach
TL;DR: In this paper, the authors present an asymptotics for orthogonal polynomials in Riemann-Hilbert problems and Jacobi operators for continued fractions.
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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.
Journal ArticleDOI
A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptotics for the MKdV equation
Percy Deift,Xin Zhou +1 more
TL;DR: In this article, the authors present an approach to analyze the asymptotics of oscillatory Riemann-Hilbert problems with respect to the modified Korteweg-de Vries (MKdV) equation.
Journal ArticleDOI
Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory
TL;DR: In this article, asymptotics for orthogonal polynomials with respect to varying exponential weights wn(x)dx = e−nV(x)-dx on the line as n ∞ were considered.