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Direct and inverse results for multipoint Hermite–Padé approximants

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Abstract

Given a system of functions \({\mathbf {f}}=(f_1,\ldots ,f_d)\) analytic on a neighborhood of some compact subset E of the complex plane with simply connected complement in the extended complex plane, we give necessary and sufficient conditions for the convergence with geometric rate of the common denominators of row sequences of multipoint Hermite–Padé approximants under a general extremal condition on the table of interpolation points. The exact rate of convergence of these denominators is provided and the rate of convergence of the simultaneous approximants is estimated. These results allow us to detect the location of the poles of the system of functions which are in some sense closest to E.

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Acknowledgements

We wish to express our gratitude toward to an anonymous referee for careful reading, helpful comments, and suggestions leading to improvements of this work. The first author thanks Assoc. Prof. Chontita Rattanakul for her invaluable guidance.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Mahidol University, Rama VI Road, Ratchathewi District, Bangkok, 10400, Thailand

    N. Bosuwan

  2. Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok, 10400, Thailand

    N. Bosuwan

  3. Department of Mathematics, Universidad Carlos III de Madrid, Avda. Universidad 30, 28911, Leganés, Madrid, Spain

    G. López Lagomasino & Y. Zaldivar Gerpe

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  1. N. Bosuwan
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  2. G. López Lagomasino
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  3. Y. Zaldivar Gerpe
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Correspondence to N. Bosuwan.

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Dedicated to Professor Stephen J. Gardiner on the occasion of his 60th birthday.

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The research of N. Bosuwan was supported by the Strengthen Research Grant for New Lecturer from the Thailand Research Fund and the Office of the Higher Education Commission (MRG6080133) and Faculty of Science, Mahidol University. The research of G. López Lagomasino and Y. Zaldivar Gerpe received support from Research Grant MTM 2015-65888-C4-2-P of Ministerio de Economía, Industria y Competitividad, Spain.

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Bosuwan, N., López Lagomasino, G. & Zaldivar Gerpe, Y. Direct and inverse results for multipoint Hermite–Padé approximants. Anal.Math.Phys. 9, 761–779 (2019). https://doi.org/10.1007/s13324-019-00316-8

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  • DOI: https://doi.org/10.1007/s13324-019-00316-8

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