On the Singularity of Multivariate Hermite Interpolation
This work provides theoretical insights into the singularity of Hermite interpolation for mathematicians working in approximation theory.
The paper studies the singularity of multivariate Hermite interpolation of type total degree, proving that all such interpolations on m = d + k points in R^d are singular if d ≥ 2k, and completely solving the problem for m ≤ d + 3 nodes. It also provides a method to compute the interpolation space.
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.