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. Work supported by the National Science Foundation under Grant DMS-0604056, by the “ex-60%” funds of the Universities of Padova and Verona, and by the INdAM-GNCS.