Open AccessAlgebro-geometric aspectsof the Christoffel-Darboux kernelsfor classical orthogonal polynomials
Author(s) -
Masanori Sawa,
Yukihiro Uchida
Publication year - 2019
Publication title -
transactions of the american mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.798
H-Index - 100
eISSN - 1088-6850
pISSN - 0002-9947
DOI - 10.1090/tran/7998
Subject(s) - mathematics , diophantine equation , orthogonal polynomials , christoffel symbols , pure mathematics , hausdorff space , algebra over a field , mathematical analysis
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.