Invariant Interpolation Space for Generalized Multivariate Birkhoff Interpolation

Abstract

In this paper, we provide a detailed exposition of a generalized multivariate Birkhoff interpolation scheme (Z,S,E) and introduce the notions of invariant interpolation space and singular interpolation space. We prove that the space PS, which is spanned by the monomial sequence S, is invariant or singular if the incidence matrix E satisfies some conditions. The advantage of our results lie in the fact that we can deduce whether PS is always proper or not for all choices of the given node set Z, just from the property of the incidence matrix E, with very low computational complexity.