Diagonal Padé approximants to hyperelliptic functions
type: Item
instance of: scholarly article
title: Diagonal Padé approximants to hyperelliptic functions
zbMATH Open document ID: 0872.41006
published in: Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série VI
publication date: 1997-09-11T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/73401
review text: This paper is a step forward in the theory of convergence of diagonal Padé approximants connected with a.o. the Baker-Gammel-Wills conjecture on the existence of infinite subsequences of diagonal approximants to meromorphic functions outside a closed disk, converging locally uniformly on sets omitting the poles. In view of the length of the paper it is not possible to cover in detail the wealth of material presented. The main subject is the study of hyperelliptic functions, analytic at infinity, with \(2m\) branch points (\(m\geq 1\)), which can be represented as \[ f(z)=r_1(z)+r_2(z)y\text{ with }y:=\sqrt{(z-a_1)\cdots(z-a_{2m})}, \] where \(r_1, r_2\) are rational functions. The main results proved in the paper concern a.o. the `recapturing' of poles \(z_0\) of \(f\) by \([n/n]\) PA's with \(n\geq n_0\), when there are so-called `spurious poles' (say \(k\)), the PA has \(k\) zeros that converge to \(z_0\), an upper bound on the number of spurious poles, an upper bound for the distance between spurious poles and the accompanying zeros from item 2, existence of subsequences of diagonal PA's having prescribed spurious poles, locally uniform convergence after `pole-clearing', proof of the Baker-Gammel-Wills conjecture. The final section gives a new proof of a result due to \textit{S. Dumas} [Sur le développement des fonctions elliptiques en fractions continues, Thésis, Univ. Zürich, (1908; FJM 39.05170.01)] on the hyperelliptic function \[ f(z)=\sqrt{(z-a_1)(_z-a_2)(z-a_3)(z-a_4)} -z^2+(a_1+a_2+a_3+a_4)z/2, \] which is analytic at infinity, the \(a_j\) are pairwise distinct.
Mathematics Subject Classification ID: 41A21
Mathematics Subject Classification ID: 41A25
Mathematics Subject Classification ID: 30E10
zbMATH DE Number: 1013859
zbMATH Keywords: rational approximation
zbMATH Keywords: Padé approximation
zbMATH Keywords: spurious poles
zbMATH Keywords: hyperelliptic functions
zbMATH Keywords: Baker-Gammel-Wills conjecture
author: Herbert Stahl
reviewed by: Marcel G. de Bruin
MaRDI profile type: Publication
OpenAlex ID: W2312486401
cites work: An investigation of the applicability of the Padé approximant method
cites work: Q3997784
cites work: Q5556574
cites work: A LOCAL CONDITION OF SINGLE-VALUEDNESS OF ANALYTIC FUNCTIONS
cites work: EQUILIBRIUM DISTRIBUTIONS AND DEGREE OF RATIONAL APPROXIMATION OF ANALYTIC FUNCTIONS
cites work: Q3285751
cites work: The ideas of P. L. Čebyšev and A. A. Markov in the theory of limiting values of integrals and their further development
cites work: Q5666837
cites work: Divergence of complex rational approximations
cites work: The convergence of Padé approximants of meromorphic functions
cites work: Asymptotics of diagonal Hermite-Padé polynomials
cites work: Über die Kapazität ebener Kontinuen
cites work: Padé approximants and convergence in capacity
cites work: ON THE CONVERGENCE OF PADÉ APPROXIMANTS IN CLASSES OF HOLOMORPHlC FUNCTIONS
cites work: Q5554461
cites work: Q3966409
cites work: Extremal domains associated with an analytic function I
cites work: The structure of extremal domains associated with an analytic function
cites work: The convergence of Padé approximants to functions with branch points
cites work: Q5289656
DOI: 10.5802/AFST.851
Recommended article: Uniform convergence of Padé diagonal approximants for hyperelliptic functions
Recommended article: The asymptotic behaviour of diagonal Padé approximants for hyperelliptic functions of genusg=2
Recommended article: On the diagonal Padé approximants of meromorphic functions
Recommended article: Q3811109
Recommended article: Padé approximants to certain elliptic-type functions
Recommended article: Padé approximations and diophantine geometry
Recommended article: Diagonal convergence of the remainder Padé approximants for the Hurwitz zeta function
Recommended article: Diophantine approximation of \(p\)-adic elliptic logarithms
Recommended article: On interpolation properties of diagonal Padé approximants of elliptic functions
Recommended article: Diophantine approximation on hyperbolic Riemann surfaces
description: scientific article; zbMATH DE number 1013859