  Hermite interpolation property is desired in applied and computational
mathematics. Hermite and vector subdivision schemes are of interest in CAGD for
generating subdivision curves and in computational mathematics for building
Hermite wavelets to numerically solve partial differential equations. In
contrast to well-studied scalar subdivision schemes, Hermite and vector
subdivision schemes employ matrix-valued masks and vector input data, which
make their analysis much more complicated and difficult than their scalar
counterparts. Despite recent progresses on Hermite subdivision schemes, several
key questions still remain unsolved, for example, characterization of Hermite
masks, factorization of matrix-valued masks, and convergence of Hermite
subdivision schemes. In this paper, we shall study Hermite subdivision schemes
through investigating vector subdivision operators acting on vector polynomials
and establishing the relations among Hermite subdivision schemes, vector
cascade algorithms and refinable vector functions. This approach allows us to
resolve several key problems on Hermite subdivision schemes including
characterization of Hermite masks, factorization of matrix-valued masks, and
convergence of Hermite subdivision schemes.
