
SIGMA 8 (2012), 067, 29 pages arXiv:1204.4501
https://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.
pdf (813 kb)
tex (857 kb)
References
 Beerends R.J., Chebyshev polynomials in several variables and the radial part
of the LaplaceBeltrami operator, Trans. Amer. Math. Soc.
328 (1991), 779814.
 Conway J.H., Sloane N.J.A., Sphere packings, lattices and groups,
Grundlehren der Mathematischen Wissenschaften, Vol. 290, 3rd ed.,
SpringerVerlag, New York, 1999.
 Dudgeon D.E., Mersereau R.M., Multidimensional digital signal processing,
PrenticeHall Inc, Englewood Cliffs, New Jersey, 1984.
 Dunkl C.F., Xu Y., Orthogonal polynomials of several variables,
Encyclopedia of Mathematics and its Applications, Vol. 81, Cambridge
University Press, Cambridge, 2001.
 Fuglede B., Commuting selfadjoint partial differential operators and a group
theoretic problem, J. Funct. Anal. 16 (1974), 101121.
 Koornwinder T.H., Orthogonal polynomials in two variables which are
eigenfunctions of two algebraically independent partial differential
operators. III, Nederl. Akad. Wetensch. Proc. Ser. A 77
(1974), 357369.
 Koornwinder T.H., Twovariable analogues of the classical orthogonal
polynomials, in Theory and Application of Special Functions (Proc.
Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison,
Wis., 1975), Academic Press, New York, 1975, 435495.
 Krall H.L., Sheffer I.M., Orthogonal polynomials in two variables, Ann.
Mat. Pura Appl. (4) 76 (1967), 325376.
 Li H., Sun J., Xu Y., Discrete Fourier analysis, cubature, and interpolation
on a hexagon and a triangle, SIAM J. Numer. Anal. 46
(2008), 16531681, arXiv:0712.3093.
 Li H., Sun J., Xu Y., Discrete Fourier analysis with lattices on planar
domains, Numer. Algorithms 55 (2010), 279300,
arXiv:0910.5286.
 Li H., Xu Y., Discrete Fourier analysis on fundamental domain and simplex of
A_{d} lattice in dvariables, J. Fourier Anal. Appl.
16 (2010), 383433, arXiv:0809.1079.
 Marks II R.J., Introduction to Shannon sampling and interpolation theory,
Springer Texts in Electrical Engineering, SpringerVerlag, New York, 1991.
 Moody R.V., Patera J., Cubature formulae for orthogonal polynomials in terms of
elements of finite order of compact simple Lie groups, Adv. in
Appl. Math. 47 (2011), 509535, arXiv:1005.2773.
 MuntheKaas H.Z., On group Fourier analysis and symmetry preserving
discretizations of PDEs, J. Phys. A: Math. Gen. 39
(2006), 55635584.
 Stroud A.H., Approximate calculation of multiple integrals, PrenticeHall
Series in Automatic Computation, PrenticeHall Inc., Englewood Cliffs, N.J.,
1971.
 Suetin P.K., Orthogonal polynomials in two variables, Analytical
Methods and Special Functions, Vol. 3, Gordon and Breach Science Publishers,
Amsterdam, 1999.
 Sun J., Multivariate Fourier series over a class of non tensorproduct
partition domains, J. Comput. Math. 21 (2003), 5362.
 Szajewska M., Four types of special functions of G_{2} and their
discretization, Integral Transforms Spec. Funct. 23 (2012),
455472, arXiv:1101.2502.
 Xu Y., Polynomial interpolation in several variables, cubature formulae, and
ideals, Adv. Comput. Math. 12 (2000), 363376.

