Cubature formula and interpolation on the cubic domain

Abstract

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 $[-1,1{]}^3$.

27 pages, 3 fugures