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The Saff–Varga Width Conjecture and Entire Functions with Simple Exponential Growth

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Constructive Approximation Aims and scope

Abstract

We show that the partial sums of the power series for a certain class of entire functions possess scaling limits in various directions in the complex plane. In doing so, we obtain information about the zeros of the partial sums. We will only assume that these entire functions have a certain asymptotic behavior at infinity. With this information, we will partially verify for this class of functions a conjecture on the location of the zeros of their partial sums known as the Saff–Varga width conjecture. Numerical results and figures are included to illustrate the results obtained for several well-known functions including the Airy functions and the parabolic cylinder functions.

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Acknowledgements

In memory of my grandmother, Jacquelyn Gray Belzano.

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Authors and Affiliations

  1. Salem, OR, USA

    Antonio R. Vargas

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  1. Antonio R. Vargas
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Correspondence to Antonio R. Vargas.

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Communicated by Doron S. Lubinsky.

This research was supported in part by an Izaak Walton Killam Memorial Scholarship and by long-term structural funding through a Methusalem grant from the Flemish government.

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Vargas, A.R. The Saff–Varga Width Conjecture and Entire Functions with Simple Exponential Growth. Constr Approx 49, 307–383 (2019). https://doi.org/10.1007/s00365-018-9422-x

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  • DOI: https://doi.org/10.1007/s00365-018-9422-x

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