Multivariate polynomial interpolation on Lissajous-Chebyshev nodes
For researchers in approximation theory and numerical analysis, this provides a theoretical generalization of known interpolation methods to higher-dimensional non-tensor product node sets.
This work develops multivariate polynomial interpolation and quadrature rules on Lissajous-Chebyshev node sets, achieving unique interpolation in spaces of multivariate Chebyshev polynomials by deriving a discrete orthogonality structure. The results generalize one-dimensional Chebyshev-Gauß-Lobatto points and two-dimensional Padua points to higher dimensions.
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-Gauß-Lobatto points in dimension one and the Padua points in dimension two.