Cubature formula and interpolation on the cubic domain

Several cubature formulas on the cubic domains are derived using the discrete Fourier analysis associated with lattice tiling, as developed in \cite{LSX}. The main results consist of a new derivation of the Gaussian type cubature for the product Chebyshev weight functions and associated interpolation polynomials on $[-1,1]^2$, as well as new results on $[-1,1]^3$. In particular, compact formulas for the fundamental interpolation polynomials are derived, based on $n^3/4 +\CO(n^2)$ nodes of a cubature formula on $[-1,1]^3$.