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

TL;DR

This paper develops discrete Fourier analysis, cubature, and interpolation methods on a hexagon and triangle, including explicit formulas and Lebesgue constant estimates, advancing trigonometric approximation techniques on these domains.

Contribution

It introduces a detailed discrete Fourier analysis on a hexagon and derives new cubature and interpolation formulas on the triangle and hypocycloid regions.

Findings

Explicit Lagrange interpolation formula with Lebesgue constant ~ (log n)^2

Gauss cubature established on the hypocycloid

Analysis on the triangle derived from hexagon analysis

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.