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

Abstract

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. The power of our factorization framework lies in the reduction of computational effort for large d: In order to prove C d -convergence, up to now, d factorization steps were needed, while our method requires only one step, independently of d. Furthermore, in this paper, we show by an example that the spectral condition is not equivalent to the reproduction of polynomials.

A note on spectral properties of Hermite subdivision operators

“…The paper [24] also shows how to relax the spectral condition, but still retain factorization and convergence results. Furthermore, we recently showed that in general the spectral condition of order d does not imply the reproduction of polynomials up to degree d of the associated scheme [28]. These rather surprising results are the motivation for this paper.…”

“…The spectral condition is an important property for the factorizability of an Hermite subdivision scheme [23,28]. Nevertheless, it is known that it is not necessary for the convergence of a scheme [24].…”

“…Theorem 9 shows that polynomial reproduction is equivalent to the spectral condition with very special spectral polynomials. Due to this reason, it can happen that the spectral condition of order ℓ is satisfied, but polynomials up to degree ℓ are not reproduced: In [28] we study the Hermite scheme H 1 (with θ = 1/32) by [20] and show that it satisfies the spectral condition of order 4 with spectral polynomials 1, x,…”

“…This is quite an interesting example. Since we have the spectral condition up to order 4, factorizations of the subdivision operator up to order 4 are possible, see [20,28]. It can even be proved that this scheme produces C 4 limits [20,28], even though it only reproduces polynomials up to degree 3.…”

“…Therefore, properties such as convergence, regularity of the limit curve, approximation order, etc. are strongly connected to properties of the subdivision operator [1,2,12,23,28,34].…”