On the Singularity of Multivariate Hermite Interpolation

TL;DR

This paper investigates the conditions under which multivariate Hermite interpolation schemes are singular or regular, providing criteria, complete solutions for small node counts, and a method to compute the interpolation space.

Contribution

It introduces a method to judge singularity, characterizes when Hermite interpolation is singular or regular, and fully solves the problem for small node counts.

Findings

All Hermite interpolations on m=d+k points in R^d are singular if d≥2k.

Complete solutions for Hermite interpolation on m≤d+3 nodes.

Identification of specific cases where regular interpolation schemes exist.

Abstract

In this paper we study the singularity of multivariate Hermite interpolation of type total degree. We present a method to judge the singularity of the interpolation scheme considered and by the method to be developed, we show that all Hermite interpolation of type total degree on $m=d+k$ points in $\R^d$ is singular if $d\geq 2k$. And then we solve the Hermite interpolation problem on $m\leq d+3$ nodes completely. Precisely, all Hermite interpolations of type total degree on $m\leq d+1$ points with $d\geq 2$ are singular; for $m=d+2$ and $m=d+3$, only three cases and one case can produce regular Hermite interpolation schemes, respectively. Besides, we also present a method to compute the interpolation space for Hermite interpolation of type total degree.