Hyperinterpolation in the cube

We construct an hyperinterpolation formula of degree n in the three-dimensional cube, by using the numerical cubature formula for the product Chebyshev measure given by the product of a (near) minimal formula in the square with Gauss-Chebyshev-Lobatto quadrature. The underlying function is sampled at N ∼ n/2 points, whereas the hyperinterpolation polynomial is determined by its (n + 1)(n + 2)(n + 3)/6 ∼ n/6 coefficients in the trivariate Chebyshev orthogonal basis. The effectiveness of the method is shown by a numerical study of the Lebesgue constant, which turns out to increase like log (n), and by the application to several test functions.