Trivariate polynomial approximation on Lissajous curves

We study Lissajous curves in the three-dimensional cube that generate algebraic cubature formulas on a special family of rank-1 Chebyshev lattices. These formulas are used to construct trivariate hyperinterpolation polynomials via a single one-dimensional Fast Chebyshev Transform (by the Chebfun package) and to compute discrete extremal sets of Fekete and Leja type for trivariate polynomial interpolation. Applications could arise in the framework of Lissajous sampling for Magnetic Particle Imaging.