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A discrete Fourier analysis on the fundamental domain $\Omega_d$ of the$d$-dimensional lattice of type $A_d$ is studied, where $\Omega_2$ is theregular hexagon and $\Omega_3$ is the rhombic dodecahedron, and analogousresults on $d$-dimensional simplex are derived by considering invariant andanti-invariant elements. Our main results include Fourier analysis intrigonometric functions, interpolation and cubature formulas on these domains.In particular, a trigonometric Lagrange interpolation on the simplex is shownto satisfy an explicit compact formula and the Lebesgue constant of theinterpolation is shown to be in the order of $(\log n)^d$. The basictrigonometric functions on the simplex can be identified with Chebyshevpolynomials in several variables already appeared in literature. We studycommon zeros of these polynomials and show that they are nodes for a family ofGaussian cubature formulas, which provides only the second known example ofsuch formulas.

Discrete Fourier analysis on fundamental domain of $A_d$ lattice and on simplex in $d$-variables