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From Operator Theory to Orthogonal Polynomials, Combinatorics, and Number Theory pp 213–238Cite as

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On Foci of Ellipses Inscribed in Cyclic Polygons

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Part of the Operator Theory: Advances and Applications book series (OT,volume 285)

Abstract

Given a natural number n ≥ 3 and two points a and b in the unit disk \(\mathbb {D}\) in the complex plane, it is known that there exists a unique elliptical disk having a and b as foci that can also be realized as the intersection of a collection of convex cyclic n-gons whose vertices fill the whole unit circle \(\mathbb {T}\). What is less clear is how to find a convenient formula or expression for such an elliptical disk. Our main results reveal how orthogonal polynomials on the unit circle provide a useful tool for finding such a formula for some values of n. The main idea is to realize the elliptical disk as the numerical range of a matrix and the problem reduces to finding the eigenvalues of that matrix.

Keywords

  • Orthogonal polynomials
  • Poncelet Ellipses
  • Blaschke products

To our friend, colleague and mentor Lance Littlejohn in celebration of his distinguished career.

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Notes

  1. 1.

    We rotate τ clockwise because of the complex conjugation in the Szegő recursion.

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Acknowledgements

Author Andrei Martinez-Finkelshtein was partially supported by Simons Foundation Collaboration Grants for Mathematicians (grant 710499) and by the Spanish Government–European Regional Development Fund (grant MTM2017-89941-P), Junta de Andalucía (research group FQM-229 and Instituto Interuniversitario Carlos I de Física Teórica y Computacional), and by the University of Almería (Campus de Excelencia Internacional del Mar CEIMAR).

Author Brian Simanek graciously acknowledges support from Simons Foundation Collaboration Grant 707882.

Author information

Authors and Affiliations

  1. Department of Mathematics, Baylor University, Waco, TX, USA

    Markus Hunziker & Brian Simanek

  2. Department of Mathematics, Baylor University, Waco, TX, USA

    Andrei Martinez-Finkelshtein

  3. Department of Mathematics, University of Almería, Almería, Spain

    Andrei Martinez-Finkelshtein

  4. Department of Mathematics, Mississippi College, Clinton, Mississippi, USA

    Taylor Poe

Authors
  1. Markus Hunziker
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  2. Andrei Martinez-Finkelshtein
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  3. Taylor Poe
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  4. Brian Simanek
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Corresponding author

Correspondence to Brian Simanek .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Baylor University, Waco, TX, USA

    Prof. Fritz Gesztesy

  2. Department of Mathematics, Baylor University, Waco, TX, USA

    Prof. Andrei Martinez-Finkelshtein

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Hunziker, M., Martinez-Finkelshtein, A., Poe, T., Simanek, B. (2021). On Foci of Ellipses Inscribed in Cyclic Polygons. In: Gesztesy, F., Martinez-Finkelshtein, A. (eds) From Operator Theory to Orthogonal Polynomials, Combinatorics, and Number Theory . Operator Theory: Advances and Applications, vol 285. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-75425-9_12

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