Discrete Fourier analysis on a dodecahedron and a tetrahedron

TL;DR

This paper develops discrete Fourier analysis techniques on dodecahedron and tetrahedron domains, deriving interpolation and cubature formulas, with explicit Lagrange interpolation and Lebesgue constant estimates.

Contribution

It introduces a novel discrete Fourier analysis framework on polyhedral domains, including explicit interpolation formulas and Lebesgue constant bounds for the tetrahedron.

Findings

Explicit trigonometric Lagrange interpolation formula

Lebesgue constant of order (log n)^3

Fourier analysis results transferred from dodecahedron to tetrahedron

Abstract

A discrete Fourier analysis on the dodecahedron is studied, from which results on a tetrahedron is deduced by invariance. The results include Fourier analysis in trigonometric functions, interpolation and cubature formulas on these domains. In particular, a trigonometric Lagrange interpolation on the tetrahedron is shown to satisfy an explicit compact formula and the Lebesgue constant of the interpolation is shown to be in the order of $(\log n)^3$.