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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Algebro-geometric aspectsof the Christoffel-Darboux kernelsfor classical orthogonal polynomials
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by Masanori Sawa and Yukihiro Uchida PDF
Trans. Amer. Math. Soc. 373 (2020), 1243-1264 Request permission

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
  • MR Author ID: 776019
  • 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
  • MR Author ID: 782178
  • Email: yuchida@tmu.ac.jp
  • 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
  • © Copyright 2019 American Mathematical Society
  • 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
  • MathSciNet review: 4068263