  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 the
regular hexagon and $\Omega_3$ is the rhombic dodecahedron, and analogous
results on $d$-dimensional simplex are derived by considering invariant and
anti-invariant elements. Our main results include Fourier analysis in
trigonometric functions, interpolation and cubature formulas on these domains.
In particular, a trigonometric Lagrange interpolation on the simplex is shown
to satisfy an explicit compact formula and the Lebesgue constant of the
interpolation is shown to be in the order of $(\log n)^d$. The basic
trigonometric functions on the simplex can be identified with Chebyshev
polynomials in several variables already appeared in literature. We study
common zeros of these polynomials and show that they are nodes for a family of
Gaussian cubature formulas, which provides only the second known example of
such formulas.
