Discrete Fourier analysis and Chebyshev polynomials with G2 group

Details Discrete Fourier analysis and Chebyshev polynomials with G2 group Discrete Fourier analysis and Chebyshev polynomials with G2 group

2012-04-19

arxiv.org/abs/1204.4501v1

Mathematics

Numerical Analysis

29 pages

Scientific paper

The discrete Fourier analysis on the $30^{\degree}$-$60^{\degree}$-$90^{\degree}$ 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 lead to a Sturm-Liouville eigenvalue problem that contains two parameters, whose solutions are analogue 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.

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