Discrete Fourier Analysis and Chebyshev Polynomials with $G_2$ Group

TL;DR

This paper develops a framework connecting discrete Fourier analysis on a specific triangle with Chebyshev polynomials derived from the $G_2$ symmetry group, leading to new orthogonal polynomials and cubature rules.

Contribution

It introduces generalized Chebyshev polynomials associated with the $G_2$ group and explores their properties, including eigenvalue problems and zero distributions, extending classical orthogonal polynomial theory.

Findings

Derived discrete Fourier analysis on a triangle from hexagon results.

Defined new families of Chebyshev polynomials with $G_2$ symmetry.

Established cubature rules of Gauss type using polynomial zeros.

Abstract

The discrete Fourier analysis on the $30^{\degree}$-$60^{\degree}$-$90^{\degree}$ 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 Sturm-Liouville eigenvalue problem that contains two parameters, whose solutions are analogues of the Jacobi polynomials. Under a concept of $m$-degree 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.