Discrete Fourier Analysis and Chebyshev Polynomials with G2 Group

• Published in: Symmetry, integrability and geometry, methods and applications Vol. 8; p. 067
• Main Authors: Li, Huiyuan, Sun, Jiachang, Xu, Yuan
• Format: Journal Article
• Language: English
• Published: Kiev National Academy of Sciences of Ukraine 2012; National Academy of Science of Ukraine
• Subjects: discrete Fourier series; Fourier analysis; group G2; orthogonal polynomials; PDE; trigonometric
• ISSN: 1815-0659; 1815-0659
• DOI: 10.3842/SIGMA.2012.067

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. [ProQuest: [...] denotes formulae omitted.]