Convergent Interpolation to Cauchy Integrals over Analytic Arcs

Baratchart, Laurent; Yattselev, Maxim L.

doi:10.1007/s10208-009-9042-8

Cited by 19 publications

(52 citation statements)

References 57 publications

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“…Finally, it is worth mentioning that in the framework of [21], sufficient conditions for existence of symmetric contours in harmonic fields were developed by Rakhmanov in [32]. Let us stress that in [32] given a harmonic field one looks for a system of arcs connecting certain points that is symmetric with respect to the field while in [9] and further below in Theorem 3.2 one starts with a system of arcs for which a measure that makes it symmetric is then produced (the corresponding field is given by the logarithmic potential of the measure).…”

Section: Stahl-gonchar-rakhmanov Theorymentioning

confidence: 99%

Symmetric contours and convergent interpolation

Yattselev

2018

Journal of Approximation Theory

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Abstract. The essence of Stahl-Gonchar-Rakhmanov theory of symmetric contours as applied to the multipoint Padé approximants is the fact that given a germ of an algebraic function and a sequence of rational interpolants with free poles of the germ, if there exists a contour that is "symmetric" with respect to the interpolation scheme, does not separate the plane, and in the complement of which the germ has a single-valued continuation with non-identically zero jump across the contour, then the interpolants converge to that continuation in logarithmic capacity in the complement of the contour. The existence of such a contour is not guaranteed. In this work we do construct a class of pairs interpolation scheme/symmetric contour with the help of hyperelliptic Riemann surfaces (following the ideas of Nuttall & Singh [28] and Baratchart & the author [9]). We consider rational interpolants with free poles of Cauchy transforms of non-vanishing complex densities on such contours under mild smoothness assumptions on the density. We utilize∂-extension of the Riemann-Hilbert technique to obtain formulae of strong asymptotics for the error of interpolation.

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Section: Stahl-gonchar-rakhmanov Theorymentioning

confidence: 99%

Symmetric contours and convergent interpolation

Yattselev

2018

Journal of Approximation Theory

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“…In both steps we proceed from the contrary: assuming that the desired assertion is not valid we come to a contradiction with orthogonality relations (11) which may be equivalently written as follows s k=1 I n,k = 0, where I n,k = I n,k (G n ) = Γ k q n,k (z)G n (z)f k (z)dz (52) and G n ∈ P ns . For the sake of convenience of the reader we will first present detail of the proof for the case s = 2.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…Many part of this theory may be generalized, one way or another, to the weighted case and, then, to the vector case s > 1 (for the weighted case [27] and also [11], [12], [17], [19], [21].) However, generalizations are not always obvious and very often exist only as conjectures; see [65], [66].…”

Section: Stahl's Theorem Extremal Compact Set γ(F )mentioning

confidence: 99%

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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“…A comprehensive account of many recent developments in the field can be found in the monograph by Simon [43]. We mention, in passing, that progress is also being made in developing the theory of non Hermitian orthogonality with respect to complex measures, which is intimately related to rational approximation and interpolation; see, for example, Aptekarev [4], Aptekarev-Van Assche [5], Baratchart-Küstner-Totik [7], and Baratchart-Yattselev [8].…”

Section: Introductionmentioning

confidence: 97%

Multipoint Schur algorithm and orthogonal rational functions, I: Convergence properties

Baratchart

Kupin

Lunot³

et al. 2011

JAMA

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Classical Schur analysis is intimately related to the theory of orthogonal polynomials on the circle. We investigate the connection between multipoint Schur analysis and orthogonal rational functions. Specifically, we study the convergence of the Wall rational functions via the development of a rational analogue to the Szegő theory in the case that interpolation points may accumulate on the unit circle. This leads to a generalization of results in [22,10], and new types of asymptotics.

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Convergent Interpolation to Cauchy Integrals over Analytic Arcs

Cited by 19 publications

References 57 publications

Symmetric contours and convergent interpolation

Symmetric contours and convergent interpolation

Zero distribution for Angelesco Hermite–Padé polynomials

Multipoint Schur algorithm and orthogonal rational functions, I: Convergence properties

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