Stirling numbers and Gregory coefficients for the factorization of Hermite subdivision operators

In this paper we present a factorization framework for Hermite subdivision schemes refining function values and first derivatives, which satisfy a spectral condition of high order. In particular we show that spectral order $d$ allows for $d$ factorizations of the subdivision operator with respect to the Gregory operators: a new sequence of operators we define using Stirling numbers and Gregory coefficients. We further prove that the $d$th factorization provides a ‘convergence from contractivity’ method for showing $C^d$-convergence of the associated Hermite subdivision scheme. Gregory operators are derived by explicitly solving a recursion based on the Taylor operator and iterated vector scheme factorizations. The explicit expression of these operators allows one to compute the $d$th factorization directly from the mask of the Hermite scheme. In particular, it is not necessary to compute intermediate factorizations, which simplifies the procedures used up to now.