Symmetric contours and convergent interpolation

JOURNAL OF APPROXIMATION THEORY

卷:225

Article

Yattselev, Maxim L.¹ 

[1] Indiana Univ Purdue Univ, Dept Math Sci, 402 North Blackford St, Indianapolis, IN 46202 USA

关键词: Multipoint Pad6 approximation;    Orthogonal polynomials;    Non-Hermitian orthogonality;    Strong asymptotics;    S-contours;    Matrix Riemann Hilbert approach;   

DOI  :  10.1016/j.jat.2017.10.003

来源: Elsevier

【 摘 要 】

The essence of Stahl-Gonchar-Rakhmanov theory of symmetric contours as applied to the multipoint Fade 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 and Singh, 1977; Baratchart and Yattselev, 2009). 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 8-extension of the Riemann-Hilbert technique to obtain formulae of strong asymptotics for the error of interpolation. (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

Free