New cubature formulae and hyperinterpolation in three variables

A new algebraic cubature formula of degree 2n+1 for the product Chebyshev measure in the d-cube with ≈n d /2 d-1 nodes is established. The new formula is then applied to polynomial hyperinterpolation of degree n in three variables, in which coefficients of the product Chebyshev orthonormal basis are computed by a fast algorithm based on the 3-dimensional FFT. Moreover, integration of the hyperinterpolant provides a new Clenshaw-Curtis type cubature formula in the 3-cube. © 2009 Springer Science + Business Media B.V.