Multivariate polynomial interpolation on Lissajous-Chebyshev nodes

TL;DR

This paper develops multivariate polynomial interpolation and quadrature rules on Lissajous-Chebyshev nodes, generalizing classical univariate and bivariate results to higher dimensions using orthogonality structures.

Contribution

It introduces a new classification of multivariate Lissajous curves and derives a discrete orthogonality structure enabling unique polynomial interpolation on these nodes.

Findings

Established a discrete orthogonality structure for Lissajous-Chebyshev nodes.

Achieved unique polynomial interpolation in multivariate Chebyshev polynomial spaces.

Generalized classical interpolation results to higher dimensions.

Abstract

In this article, we study multivariate polynomial interpolation and quadrature rules on non-tensor product node sets related to Lissajous curves and Chebyshev varieties. After classifying multivariate Lissajous curves and the interpolation nodes linked to these curves, we derive a discrete orthogonality structure on these node sets. Using this orthogonality structure, we obtain unique polynomial interpolation in appropriately defined spaces of multivariate Chebyshev polynomials. Our results generalize corresponding interpolation and quadrature results for the Chebyshev-Gau{\ss}-Lobatto points in dimension one and the Padua points in dimension two.