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

Cubature formula and interpolation on the cubic domain