  Due to properties such as interpolation, smoothness, and spline connections,
Hermite subdivision schemes employ fast iterative algorithms for geometrically
modeling curves/surfaces in CAGD and for building Hermite wavelets in numerical
PDEs. In this paper we introduce a notion of generalized Hermite (dyadic)
subdivision schemes and then we characterize their convergence, smoothness and
underlying matrix masks with or without interpolation properties. We also
introduce the notion of linear-phase moments for achieving the
polynomial-interpolation property. For any given positive integer m, we
constructively prove that there always exist convergent smooth generalized
Hermite subdivision schemes with linear-phase moments such that their basis
vector functions are spline functions in $C^m$ and have linearly independent
integer shifts. As byproducts, our results resolve convergence, smoothness and
existence of Lagrange, Hermite, or Birkhoff subdivision schemes. Even in
dimension one our results significantly generalize and extend many known
results on extensively studied univariate Hermite subdivision schemes. To
illustrate the theoretical results in this paper, we provide examples of
convergent generalized Hermite subdivision schemes with symmetric matrix masks
having short support and smooth basis vector functions with or without
interpolation property.
