Hyperinterpolation in the cube

We construct an hyperinterpolation formula of degree nn 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∼n3/2N∼n3/2 points, whereas the hyperinterpolation polynomial is determined by its (n+1)(n+2)(n+3)/6∼n3/6(n+1)(n+2)(n+3)/6∼n3/6 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 log3(n)log3(n), and by the application to several test functions.