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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Algebro-geometric aspectsof the Christoffel-Darboux kernelsfor classical orthogonal polynomials


Authors: Masanori Sawa and Yukihiro Uchida
Journal: Trans. Amer. Math. Soc. 373 (2020), 1243-1264
MSC (2010): Primary 05E99, 33C45, 65D32; Secondary 11D72
DOI: https://doi.org/10.1090/tran/7998
Published electronically: November 5, 2019
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Abstract: In this paper we study algebro-geometric aspects of the Christoffel-Darboux kernels for classical orthogonal polynomials with rational coefficients. We find a novel connection between a projective curve defined by theChristoffel-Darboux kernel and a system of Diophantine equations, which was originally designed by Hausdorff (1909) for applications to Waring's problem, and which is closely related to quadrature formulas in numerical analysis and Gaussian designs in algebraic combinatorics. We prove some nonsolvability results of such Hausdorff-type equations.


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Additional Information

Masanori Sawa
Affiliation: Graduate School of System Informatics, Kobe University, 1-1 Rokkodai, Nada, Kobe 657-8501, Japan
Email: sawa@people.kobe-u.ac.jp

Yukihiro Uchida
Affiliation: Department of Mathematical Sciences, Graduate School of Science, Tokyo Metropolitan University, 1-1 Minami-Osawa, Hachioji, Tokyo 192-0397, Japan
Email: yuchida@tmu.ac.jp

DOI: https://doi.org/10.1090/tran/7998
Keywords: Classical orthogonal polynomial, Gaussian design, Hausdorff-type equation, Christoffel-Darboux kernel, projective curve, quadrature formula, quasi-orthogonal polynomial, reproducing kernel
Received by editor(s): December 25, 2018
Received by editor(s) in revised form: July 8, 2019
Published electronically: November 5, 2019
Additional Notes: The first author was supported in part by Grant-in-Aid for Scientific Research (C) 18K03414 and Grant-in-Aid for Scientific Research (B) 15H03636 by the Japan Society for the Promotion of Science (JSPS)
The second author was also supported by Grant-in-Aid for Young Scientists (B) 25800023 by JSPS
Article copyright: © Copyright 2019 American Mathematical Society