Convergence analysis of Hermite subdivision schemes of any arity

In this paper, we give a framework to analyze the convergence and smoothness of Hermite subdivision schemes of any arity. The convergence analysis is based on the connections among Hermite subdivision schemes, vector subdivision schemes and refinable function vectors. We first investigate the relation between sum rules and vector subdivision operators acting on vector polynomials of any arity. To study Hermite subdivision schemes, then we depict all Hermite masks of convergent Hermite subdivision schemes of any arity. Under the normal form of matrix-valued masks, factorizations of Hermite masks of Hermite subdivision schemes with any arity are given. Through a quantity defined by sum rules, we can further estimate the smoothness of Hermite subdivision schemes of any arity. Moreover, we can construct Hermite subdivision schemes with arbitrarily high smoothness. Finally, some numerical examples are shown to demonstrate the theoretical results.