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Weighted Fekete Points on the Real Line and the Unit Circle

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Abstract

Weighted Fekete points are defined as those that maximize the weighted version of the Vandermonde determinant over a fixed set. They can also be viewed as the equilibrium distribution of the unit discrete charges in an external electrostatic field. While these points have many applications, they are very difficult to find explicitly, and are only known in a few (unweighted) classical cases. We give two rare explicit examples of weighted Fekete points. The first one is for the weights \(w(x)=|x-ai|^{-s}\) on the real line, with \(s\ge 1\) and \(a\ne 0,\) while the second is for the weights \(w(z)=1/|z-b|\) on the unit circle, with \(b\in {{\mathbb {R}}},\ b\ne \pm 1.\) In both cases, we provide solutions of the continuous energy problems with external fields that express the limit versions of considered weighted Fekete points problems.

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Acknowledgements

The research of the first named author has received funding from European Social Fund (Project No. 09.3.3-LMT-K-712-01-0037) under grant agreement with the Research Council of Lithuania (LMTLT). Research of the second author was partially supported by NSF via the American Institute of Mathematics, and by the Vaughn Foundation endowed Professorship in Number Theory.

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Authors and Affiliations

  1. Institute of Mathematics, Faculty of Mathematics and Informatics, Vilnius University, Naugarduko 24, 03225, Vilnius, Lithuania

    Artūras Dubickas

  2. Department of Mathematics, Oklahoma State University, Stillwater, OK, 74078, USA

    Igor Pritsker

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  1. Artūras Dubickas
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  2. Igor Pritsker
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Correspondence to Igor Pritsker.

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Communicated by Vladimir V. Andrievskii.

In memory of Prof. Dr. Stephan Ruscheweyh.

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Dubickas, A., Pritsker, I. Weighted Fekete Points on the Real Line and the Unit Circle. Comput. Methods Funct. Theory 20, 403–429 (2020). https://doi.org/10.1007/s40315-020-00317-w

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