  A new algebraic cubature formula of degree $2n+1$ for the product Chebyshev
measure in the $d$-cube with $\approx n^d/2^{d-1}$ 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.
