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International Mathematics Research Papers (2008) Vol. 2008 : article ID rpm007, 128 pages, doi:10.1093/imrp/rpm007 published on January 29, 2008
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© The Author 2008. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org.

Asymptotics of Hermite–Padé Rational Approximants for Two Analytic Functions with Separated Pairs of Branch Points (Case of Genus 0)

Alexander I. Aptekarev1, Arno B. J. Kuijlaars2 and Walter Van Assche2

1 Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Miusskaya Square 4, 125047 Moscow, Russian Federation
2 Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B box 2400, BE-3001 Leuven, Belgium

Correspondence: Correspondence to be sent to: Walter Van Assche, Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B box 2400, BE-3001 Leuven, Belgium. e-mail: walter{at}wis.kuleuven.be

We investigate the asymptotic behavior for type II Hermite–Padé approximation to two functions, where each function has two branch points and the pairs of branch points are separated. We give a classification of the cases such that the limiting counting measures for the poles of the Hermite–Padé approximants are described by an algebraic function h of order and genus 0. This situation gives rise to a vector-potential equilibrium problem for measures {lambda}, µ1, and µ2, and the poles of the common denominator are asymptotically distributed like {lambda}/2. We also work out the strong asymptotics for the corresponding Hermite–Padé approximants by using a 3 x 3 Riemann–Hilbert problem that characterizes this Hermite–Padé approximation problem.



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This Article
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