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Mathematics > Numerical Analysis

arXiv:0712.3093 (math)
[Submitted on 19 Dec 2007]

Title:Discrete Fourier analysis, Cubature and Interpolation on a Hexagon and a Triangle

Authors:Huiyuan Li, Jiachang Sun, Yuan Xu
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Abstract: 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 planer 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.
Comments: 29 pages
Subjects: Numerical Analysis (math.NA); Classical Analysis and ODEs (math.CA)
MSC classes: 41A05, 41A10
Cite as: arXiv:0712.3093 [math.NA]
  (or arXiv:0712.3093v1 [math.NA] for this version)
  https://doi.org/10.48550/arXiv.0712.3093

Submission history

From: Yuan Xu [view email]
[v1] Wed, 19 Dec 2007 04:00:14 UTC (165 KB)
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