Discrete Fourier Analysis, Cubature, and Interpolation on a Hexagon and a Triangle

Several problems of trigonometric approximation on a hexagon and a triangle are studied using the discrete Fourier transform and orthogonal polynomials of two variables. A discrete Fourier analysis on the regular hexagon is developed in detail, from which the analysis on the triangle is deduced. The results include cubature formulas and interpolation on these domains. In particular, a trigonometric Lagrange interpolation on a triangle is shown to satisfy an explicit compact formula, which is equivalent to the polynomial interpolation on a planar region bounded by Steiner's hypocycloid. The Lebesgue constant of the interpolation is shown to be in the order of $(\log n)^2$. Furthermore, a Gauss cubature is established on the hypocycloid.

[1]  T. Koornwinder Two-Variable Analogues of the Classical Orthogonal Polynomials , 1975 .

[2]  Mihail N. Kolountzakis The Study of Translational Tiling with Fourier Analysis , 2004 .

[3]  R. Marks Introduction to Shannon Sampling and Interpolation Theory , 1990 .

[4]  A. Stroud Approximate calculation of multiple integrals , 1973 .

[5]  Brian J. McCartin,et al.  Eigenstructure of the Equilateral Triangle, Part I: The Dirichlet Problem , 2003, SIAM Rev..

[6]  Yuan Xu,et al.  Orthogonal Polynomials of Several Variables , 2014, 1701.02709.

[7]  W. Fischer,et al.  Sphere Packings, Lattices and Groups , 1990 .

[8]  Huiyuan Li,et al.  Generalized Fourier transform on an arbitrary triangular domain , 2005, Adv. Comput. Math..

[9]  A.K. Krishnamurthy,et al.  Multidimensional digital signal processing , 1985, Proceedings of the IEEE.

[10]  Tom H. Koornwinder,et al.  Orthogonal polynomials in two variables which are eigenfunctions of two algebraically independent partial differential operators. III , 1974 .

[11]  Thomas C. Hales Sphere packings, I , 1997, Discret. Comput. Geom..

[12]  T. J. Rivlin An Introduction to the Approximation of Functions , 2003 .

[13]  P. Heywood Trigonometric Series , 1968, Nature.

[14]  Jia-changSun MULTIVARIATE FOURIER SERIES OVER A CLASS OF NON TENSOR-PRODUCT PARTITION DOMAINS , 2003 .

[15]  Rene F. Swarttouw,et al.  Orthogonal polynomials , 2020, NIST Handbook of Mathematical Functions.

[16]  Yuan Xu Polynomial interpolation in several variables, cubature formulae, and ideals[*]Supported by the National Science Foundation under Grant DMS-9802265. , 2000, Adv. Comput. Math..

[17]  Mark A. Pinsky,et al.  Completeness of the Eigenfunctions of the Equilateral Triangle , 1985 .

[18]  Wolfgang Ebeling,et al.  Lattices and Codes: A Course Partially Based on Lectures by Friedrich Hirzebruch , 1994 .

[19]  Bent Fuglede,et al.  Commuting self-adjoint partial differential operators and a group theoretic problem , 1974 .

[20]  Brian J. McCartin,et al.  Eigenstructure of the equilateral triangle , 2003 .

[21]  Ian H. Sloan,et al.  Multiple integration over bounded and unbounded regions , 1987 .

[22]  Brian J. McCartin,et al.  Eigenstructure of the equilateral triangle, Part II: The Neumann problem , 2002 .

[23]  R. L. Stens,et al.  Sampling theory in Fourier and signal analysis : advanced topics , 1999 .

[24]  J. R. Higgins Sampling theory in Fourier and signal analysis : foundations , 1996 .

[25]  Mark A. Pinsky,et al.  The Eigenvalues of an Equilateral Triangle , 1980 .

[26]  Yuan Xu,et al.  On bivariate Gaussian cubature formulae , 1994 .