  Employing a matrix mask, a vector subdivision scheme is a fast iterative
averaging algorithm to compute refinable vector functions for wavelet methods
in numerical PDEs and to produce smooth curves in CAGD. In sharp contrast to
the well-studied scalar subdivision schemes, vector subdivision schemes are
much less well understood, e.g., Lagrange and (generalized) Hermite subdivision
schemes are the only studied vector subdivision schemes in the literature.
Because many wavelets used in numerical PDEs are derived from refinable vector
functions whose matrix masks are not from Hermite subdivision schemes, it is
necessary to introduce and study vector subdivision schemes for any general
matrix masks in order to compute wavelets and refinable vector functions
efficiently. For a general matrix mask, we show that there is only one
meaningful way of defining a vector subdivision scheme. Motivated by vector
cascade algorithms and recent study on Hermite subdivision schemes, we shall
define a vector subdivision scheme for any arbitrary matrix mask and then we
prove that the convergence of the newly defined vector subdivision scheme is
equivalent to the convergence of its associated vector cascade algorithm. We
also study convergence rates of vector subdivision schemes. The results of this
paper not only bridge the gaps and establish intrinsic links between vector
subdivision schemes and vector cascade algorithms but also strengthen and
generalize current known results on Lagrange and (generalized) Hermite
subdivision schemes. Several examples are provided to illustrate the results in
this paper on various types of vector subdivision schemes with convergence
rates.
