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Mathematics of Computation

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Exact cubature rules for symmetric functions

Authors: J. F. van Diejen and E. Emsiz
Journal: Math. Comp. 88 (2019), 1229-1249
MSC (2010): Primary 65D32; Secondary 05E05, 15B52, 33C52, 33D52, 65T40
Published electronically: October 2, 2018
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Abstract: We employ a multivariate extension of the Gauss quadrature formula, originally due to Berens, Schmid, and Xu [Arch. Math. (Basel) 64 (1995), pp. 26-32], so as to derive cubature rules for the integration of symmetric functions over hypercubes (or infinite limiting degenerations thereof) with respect to the densities of unitary random matrix ensembles. Our main application concerns the explicit implementation of a class of cubature rules associated with the Bernstein-Szegö polynomials, which permit the exact integration of symmetric rational functions with prescribed poles at coordinate hyperplanes against unitary circular Jacobi distributions stemming from the Haar measures on the symplectic and the orthogonal groups.

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Additional Information

J. F. van Diejen
Affiliation: Instituto de Matemática y Física, Universidad de Talca, Casilla 747, Talca, Chile

E. Emsiz
Affiliation: Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Casilla 306, Correo 22, Santiago, Chile

Keywords: Cubature rules, symmetric functions, generalized Schur polynomials, Bernstein-Szeg\"o polynomials, unitary random matrix ensembles, Jacobi distributions
Received by editor(s): November 13, 2017
Received by editor(s) in revised form: April 5, 2018
Published electronically: October 2, 2018
Additional Notes: This work was supported in part by the Fondo Nacional de Desarrollo Científico y Tecnológico (FONDECYT) Grants #1170179 and #1181046.
Article copyright: © Copyright 2018 American Mathematical Society