
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 G_{2} 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 974031222, 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 G_{2}, which leads to the definition of four
families generalized Chebyshev polynomials. The study of these polynomials
leads to a SturmLiouville eigenvalue problem that contains two parameters, whose
solutions are analogues of the Jacobi polynomials. Under a concept of mdegree
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 G_{2}; PDE; orthogonal polynomials.
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