Trivariate polynomial approximation on Lissajous curves
Provides a new computational framework for multivariate polynomial approximation on specific curves, but the impact is limited to a niche domain (MPI) and the results are theoretical without concrete performance numbers.
The paper develops trivariate polynomial approximation on Lissajous curves, enabling hyperinterpolation via a single 1-D Fast Chebyshev Transform and computation of Fekete/Leja points, with potential applications in Magnetic Particle Imaging.
We study Lissajous curves in the 3-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 1-d 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 MPI (Magnetic Particle Imaging).