Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 8 (2012), 067, 29 pages      arXiv:1204.4501      http://dx.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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