О сходимости $m$-точечных аппроксимаций Паде набора многозначных аналитических функций

Buslaev, V. S.

doi:10.4213/sm8428

Cited by 30 publications

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References 13 publications

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“…There are many applications of the theorem (see e.g. reviews [6], [57] and recent papers [56], [18], [69]). In particular, after some modifications GRS-theorem may be applied to study Angelesco case of Hermite-Padé polynomials and this is a main point of this paper.…”

Section: Methodsmentioning

confidence: 99%

“…We note that in the case of multipoint Padé approximants the complement to the extremal compact set is not necessary a domain, i.e., it might consists of several domains; see [16], [17], [18]. In the case of Hermite-Padé polynomials for the general complex Nikishin systems the situation is very similar to the multipoint Padé approximants; see [43], [56], [68].…”

Section: The Grs-theoremmentioning

confidence: 99%

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Zero distribution for Angelesco Hermite–Padé polynomials

Rakhmanov¹

2018

Russ. Math. Surv.

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We consider the problem of zero distribution of the first kind Hermite-Padé polynomials associated with a vector function f = (f 1 , . . . , f s ) whose components f k are functions with a finite number of branch points in plane. We assume that branch sets of component functions are well enough separated (which constitute the Angelesco case). Under this condition we prove a theorem on limit zero distribution for such polynomials. The limit measures are defined in terms of a known vector equilibrium problem.Proof of the theorem is based on the methods developed by H. Stahl [59]-[63], A. A. Gonchar and the author [27], [55]. These methods obtained some further generalization in the paper in application to systems of polynomials defined by systems of complex orthogonality relations.Together with the characterization of the limit zero distributions of Hermite-Padé polynomials by a vector equilibrium problem we consider an alternative characterization using a Riemann surface R( f ) associated with f . In this terms we present a more general (without Angelesco condition) conjecture on the zero distribution of Hermite-Padé polynomials.Bibliography: 72 items.

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Section: Methodsmentioning

confidence: 99%

Section: The Grs-theoremmentioning

confidence: 99%

Zero distribution for Angelesco Hermite–Padé polynomials

Rakhmanov¹

2018

Russ. Math. Surv.

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On the existence of compacta of minimal capacity in the theory of rational approximation of multi-valued analytic functions

Buslaev

Suetin

2016

Journal of Approximation Theory

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АннотацияFor an interval E = [a, b] on the real line, let µ be either the equilibrium measure, or the normalized Lebesgue measure of E, and let V µ denote the associated logarithmic potential. In the present paper, we construct a function f which is analytic on E and possesses four branch points of second order outside of E such that the family of the admissible compacta of f has no minimizing elements with regard to the extremal theoretic-potential problem, in the external field equals V −µ .Bibliography: 35 items.rational approximants, Padé approximation, orthogonal polynomials, distribution of poles, convergence in capacity Dedicated to the memory of Andrei Aleksandrovich Gonchar and Herbert Stahl NotationsThroughout the paper, we use the following notations.

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On a lower bound for the rate of convergence of multipoint Padé approximants of piecewise analytic functions

Buslaev¹

2021

Izv. Math.

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О сходимости $m$-точечных аппроксимаций Паде набора многозначных аналитических функций

Cited by 30 publications

References 13 publications

Zero distribution for Angelesco Hermite–Padé polynomials

Zero distribution for Angelesco Hermite–Padé polynomials

On the existence of compacta of minimal capacity in the theory of rational approximation of multi-valued analytic functions

On a lower bound for the rate of convergence of multipoint Padé approximants of piecewise analytic functions

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