Erschienen:
Springer Science and Business Media LLC, 2021
Erschienen in:
Constructive Approximation, 53 (2021) 3, Seite 441-478
Sprache:
Englisch
DOI:
10.1007/s00365-020-09515-0
ISSN:
1432-0940;
0176-4276
Entstehung:
Anmerkungen:
Beschreibung:
<jats:title>Abstract</jats:title><jats:p>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 <jats:inline-formula><jats:alternatives><jats:tex-math>$$C_n^{(1+\alpha )}(z)$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
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</mml:math></jats:alternatives></jats:inline-formula> for <jats:inline-formula><jats:alternatives><jats:tex-math>$$\alpha >-1$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
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</mml:math></jats:alternatives></jats:inline-formula> containing the Legendre polynomials <jats:inline-formula><jats:alternatives><jats:tex-math>$$P_n(z)$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
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</mml:math></jats:alternatives></jats:inline-formula> and the subset <jats:inline-formula><jats:alternatives><jats:tex-math>$$P_n^{(\alpha +\frac{1}{2},\pm \frac{1}{2})}(z)$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
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</mml:math></jats:alternatives></jats:inline-formula> 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 <jats:inline-formula><jats:alternatives><jats:tex-math>$$\alpha \rightarrow \infty $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
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</mml:math></jats:alternatives></jats:inline-formula> 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.</jats:p>