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 [10]. 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 + O(n(2)) nodes of a cubature formula on [-1, 1](3).