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SIGMA 8 (2012), 067, 29 pages arXiv:1204.4501
https://doi.org/10.3842/SIGMA.2012.067
Discrete Fourier Analysis and Chebyshev Polynomials with G2 Group
Huiyuan Li a, Jiachang Sun a and Yuan Xu b
a) Institute of Software, Chinese Academy of Sciences, Beijing 100190, China
b) Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222, USA
Received May 04, 2012, in final form September 06, 2012; Published online October 03, 2012
Abstract
The discrete Fourier analysis on the 30°-60°-90° triangle
is deduced from the corresponding results on the regular hexagon by considering
functions invariant under the group G2, which leads to the definition of four
families generalized Chebyshev polynomials. The study of these polynomials
leads to a Sturm-Liouville eigenvalue problem that contains two parameters, whose
solutions are analogues of the Jacobi polynomials. Under a concept of m-degree
and by introducing a new ordering among monomials, these polynomials are
shown to share properties of the ordinary orthogonal polynomials. In
particular, their common zeros generate cubature rules of Gauss type.
Key words:
discrete Fourier series; trigonometric; group G2; PDE; orthogonal polynomials.
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