Trivariate polynomial approximation on Lissajous curves

TL;DR

This paper develops methods for trivariate polynomial approximation on Lissajous curves in three dimensions, enabling efficient interpolation and extremal set computation with potential applications in Magnetic Particle Imaging.

Contribution

It introduces algebraic cubature formulas on Chebyshev lattices generated by Lissajous curves and applies them to hyperinterpolation and extremal set computation in three variables.

Findings

Constructed algebraic cubature formulas on Chebyshev lattices.

Implemented trivariate hyperinterpolation using a single 1-D Fast Chebyshev Transform.

Computed discrete extremal sets of Fekete and Leja type for 3D polynomial interpolation.

Abstract

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).