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This article is cited in 59 scientific papers (total in 60 papers)
Systems of Markov functions generated by graphs and the asymptotics of their Hermite-Padé approximants
A. I. Aptekarev, V. G. Lysov M. V. Keldysh Institute for Applied Mathematics, Russian Academy of Sciences
Abstract:
The paper considers Hermite-Padé approximants to systems of Markov functions defined by means of directed graphs. The minimization problem for the energy functional is investigated for a vector measure whose components are related by a given interaction matrix and supported in some fixed system of intervals. The weak asymptotics of the approximants are obtained in terms of the solution of this problem. The defining graph is allowed to contain undirected cycles, so the minimization problem in question is considered within the class
of measures whose masses are not fixed, but allowed to ‘flow’ between intervals. Strong asymptotic formulae are also obtained. The basic tool that is used is an algebraic Riemann surface defined by means of the supports of the components of the extremal measure. The strong asymptotic formulae involve standard functions on this Riemann surface and solutions of some boundary value problems on it. The proof depends upon an asymptotic solution of the corresponding matrix Riemann-Hilbert problem.
Bibliography: 40 titles.
Keywords:
Hermite-Padé approximants, multiple orthogonal polynomials, weak and strong asymptotics, extremal equilibrium problems for a system of measures, matrix Riemann-Hilbert problem.
Received: 22.12.2008 and 03.09.2009
Citation:
A. I. Aptekarev, V. G. Lysov, “Systems of Markov functions generated by graphs and the asymptotics of their Hermite-Padé approximants”, Mat. Sb., 201:2 (2010), 29–78; Sb. Math., 201:2 (2010), 183–234
Linking options:
https://www.mathnet.ru/eng/sm7515https://doi.org/10.1070/SM2010v201n02ABEH004070 https://www.mathnet.ru/eng/sm/v201/i2/p29
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Abstract page: | 1254 | Russian version PDF: | 388 | English version PDF: | 17 | References: | 85 | First page: | 21 |
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