Abstract
We study the singularity of multivariate Hermite interpolation of type total degree onmnodes with3+d<m≤d(d+3)/2. We first check the number of the interpolation conditions and the dimension of interpolation space. And then the singularity of the interpolation schemes is decided for most cases. Also some regular interpolation schemes are derived, a few of which are proved due to theoretical argument and most of which are verified by numerical method. There are some schemes to be decided and left open.
Highlights
Let Πd be the space of all polynomials in d variables, and let Πdn be the subspace of polynomials of total degree at most n
We study the singularity of multivariate Hermite interpolation of type total degree on m nodes with 3 + d < m ≤ d(d + 3)/2
We consider the singular problem of multivariate Hermite interpolation of total degree
Summary
Let Πd be the space of all polynomials in d variables, and let Πdn be the subspace of polynomials of total degree at most n. The Hermite interpolation problem to be considered in this paper is described as follows: Find a (unique) polynomial f ∈ Πdn satisfying. Following [1, 2], such kind of problem is called Hermite interpolation of type total degree. Authors [5] made further development and gave complete description for the regularity of the interpolation problem on m = d + k (k ≤ 3) nodes, which is an extension of the results mentioned in [1, 2]. This paper is an extension of [5] and we will investigate the singularity of Hermite interpolation for m = d + k ≤ d(d + 3)/2 with d ≥ 3, k ≥ 4.
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