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 dth factorization provides a ‘convergence from contractivity’ method for showing Cd-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 dth 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.