A unifying theory for multivariate polynomial interpolation on general Lissajous-Chebyshev nodes
Provides a theoretical unification for researchers working on multivariate polynomial interpolation, but the contribution is primarily theoretical and incremental.
This paper unifies several non-tensorial polynomial interpolation schemes (Padua, Morrow-Patterson-Xu, Lissajous nodes) into a single framework using general Lissajous-Chebyshev points, deriving discrete orthogonality and efficient interpolation. No concrete numerical results are provided.
The goal of this article is to provide a general multivariate framework that synthesizes well-known non-tensorial polnomial interpolation schemes on the Padua points, the Morrow-Patterson-Xu points and the Lissajous node points into a single unified theory. The interpolation nodes of these schemes are special cases of the general Lissajous-Chebyshev points studied in this article. We will characterize these Lissajous-Chebyshev points in terms of Lissajous curves and Chebyshev varieties and derive a general discrete orthogonality structure related to these points. This discrete orthogonality is used as the key for the proof of the uniqueness of the polynomial interpolation and the derivation of a quadrature rule on these node sets. Finally, we give an efficient scheme to compute the polynomial interpolants.