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Discrete Fourier analysis and Chebyshev polynomials with $G_2$ group
Symmetry, integrability and geometry: methods and applications, Tome 8 (2012) Cet article a éte moissonné depuis la source Math-Net.Ru

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The discrete Fourier analysis on the $30^{\circ}$–$60^{\circ}$–$90^{\circ}$ triangle is deduced from the corresponding results on the regular hexagon by considering functions invariant under the group $G_2$, 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.
Keywords: discrete Fourier series; trigonometric; group $G_2$; PDE; orthogonal polynomials. 
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Huiyuan Li; Jiachang Sun; Yuan Xu. Discrete Fourier analysis and Chebyshev polynomials with $G_2$ group. Symmetry, integrability and geometry: methods and applications, Tome 8 (2012). http://geodesic.mathdoc.fr/item/SIGMA_2012_8_a66/

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