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The Jentzsch-Szegő Theorem and Balayage Measures

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Abstract

The numerous generalizations of the Jentzsch-Szegő theorem on the location of zeros of Taylor polynomials have been based so far on the extremal properties satisfied by the corresponding approximants. We do away with those kinds of assumptions and prove the theorem for a general class of interpolating polynomials. This is possible thanks to the discovery presented here that the limit distribution of the zeros of the interpolants is governed by a balayage measure depending on the distribution of the interpolation points and the region of analyticity of the function being approximated.

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Acknowledgments

The author would like to thank an anonymous referee for the remark on best approximations that appears in the penultimate paragraph of Sect. 1. This work was supported by Dirección General de Investigación, Ministerio de Educación y Ciencia, under Grant MTM2009-14668-C02-02 and by Universidad Politécnica de Madrid through Research Group “Constructive Approximation Theory and Applications.”

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  1. Departamento de Matemática Aplicada, E.T.S. Ingenieros Industriales, Universidad Politécnica de Madrid, José G. Abascal 2, 28006 , Madrid, Spain

    Bernardo de la Calle Ysern

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  1. Bernardo de la Calle Ysern
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Correspondence to Bernardo de la Calle Ysern.

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Communicated by Edward B. Saff.

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de la Calle Ysern, B. The Jentzsch-Szegő Theorem and Balayage Measures. Constr Approx 40, 307–327 (2014). https://doi.org/10.1007/s00365-014-9240-8

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  • DOI: https://doi.org/10.1007/s00365-014-9240-8

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