This article is cited in 10 papers

Discrete Fourier analysis and Chebyshev polynomials with $G_2$ group

Huiyuan Li^a, Jiachang Sun^a, Yuan Xu<sup>b

a</sup> Institute of Software, Chinese Academy of Sciences, Beijing 100190, China
^b Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222, USA

Abstract: 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.

MSC: 41A05; 41A10

Received: May 4, 2012; in final form September 6, 2012; Published online October 3, 2012

Language: English

DOI: 10.3842/SIGMA.2012.067

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ArXiv: 1204.4501