Gegenbauer and Other Planar Orthogonal Polynomials on an Ellipse in the Complex Plane

Akemann G, Nagao T, Parra Ferrada I, Vernizzi G (2020)
Constructive Approximation.

Zeitschriftenaufsatz | E-Veröff. vor dem Druck | Englisch
 
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Autor*in
Akemann, GernotUniBi; Nagao, Taro; Parra Ferrada, IvanUniBi; Vernizzi, Graziano
Abstract / Bemerkung
We show that several families of classical orthogonal polynomials on the real line are also orthogonal on the interior of an ellipse in the complex plane, subject to a weighted planar Lebesgue measure. In particular these include Gegenbauer polynomials C-n((1+alpha))(z) for alpha > -1 containing the Legendre polynomials P-n(z) and the subset P-n((alpha+12,+/- 12))(z) of the Jacobi polynomials. These polynomials provide an orthonormal basis and the corresponding weighted Bergman space forms a complete metric space. This leads to a certain family of Selberg integrals in the complex plane. We recover the known orthogonality of Chebyshev polynomials of the first up to fourth kind. The limit alpha -> infinity leads back to the known Hermite polynomials orthogonal in the entire complex plane. When the ellipse degenerates to a circle we obtain the weight function and monomials known from the determinantal point process of the ensemble of truncated unitary random matrices.
Stichworte
Planar orthogonal polynomials; Ellipse; Bergman space; Selberg integral
Erscheinungsjahr
2020
Zeitschriftentitel
Constructive Approximation
ISSN
0176-4276
eISSN
1432-0940
Page URI
https://pub.uni-bielefeld.de/record/2946498

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Akemann G, Nagao T, Parra Ferrada I, Vernizzi G. Gegenbauer and Other Planar Orthogonal Polynomials on an Ellipse in the Complex Plane. Constructive Approximation. 2020.
Akemann, G., Nagao, T., Parra Ferrada, I., & Vernizzi, G. (2020). Gegenbauer and Other Planar Orthogonal Polynomials on an Ellipse in the Complex Plane. Constructive Approximation. doi:10.1007/s00365-020-09515-0
Akemann, G., Nagao, T., Parra Ferrada, I., and Vernizzi, G. (2020). Gegenbauer and Other Planar Orthogonal Polynomials on an Ellipse in the Complex Plane. Constructive Approximation.
Akemann, G., et al., 2020. Gegenbauer and Other Planar Orthogonal Polynomials on an Ellipse in the Complex Plane. Constructive Approximation.
G. Akemann, et al., “Gegenbauer and Other Planar Orthogonal Polynomials on an Ellipse in the Complex Plane”, Constructive Approximation, 2020.
Akemann, G., Nagao, T., Parra Ferrada, I., Vernizzi, G.: Gegenbauer and Other Planar Orthogonal Polynomials on an Ellipse in the Complex Plane. Constructive Approximation. (2020).
Akemann, Gernot, Nagao, Taro, Parra Ferrada, Ivan, and Vernizzi, Graziano. “Gegenbauer and Other Planar Orthogonal Polynomials on an Ellipse in the Complex Plane”. Constructive Approximation (2020).

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arXiv: 1905.02397

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