Hierarchical Computation of Hermite Spherical Interpolant

In this paper, we propose to extend the hierarchical bivariateHermite Interpolant to the spherical case. Let $T$ be an arbitraryspherical triangle of the unit sphere $S$ and let $u$ be a functiondefined over the triangle $T$. For $k\in \mathbb{N}$, we consider aHermite spherical Interpolant problem $H_k$ defined by some datascheme $\mathcal{D}_k(u)$ and which admits a unique solution $p_k$in the space $B_{n_k}(T)$ of homogeneous Bernstein-B\''ezierpolynomials of degree $n_k=2k$ (resp. $n_k=2k+1$) defined on $T$. Wediscuss the case when the data scheme $\mathcal{D}_{r}(u)$ arenested, i.e., $\mathcal{D}_{r-1}(u)\subset \mathcal{D}_{r}(u)$ forall $1 \leq r \leq k$. This, give a recursive formulae to computethe polynomial $p_k$. Moreover, this decomposition give a new basisfor the space $B_{n_k}(T)$, which are the hierarchical structure.The method is illustrated by a simple numerical example.