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

TL;DR

This paper introduces a novel factorization framework for Hermite subdivision schemes using Stirling numbers and Gregory coefficients, significantly reducing computational effort for convergence proofs.

Contribution

It develops a new factorization approach based on Gregory operators, enabling single-step convergence proofs for high-order schemes, unlike previous multi-step methods.

Findings

The new framework reduces the number of factorization steps needed for convergence from d to 1.

Spectral order d allows for d factorizations using Gregory operators.

An example shows spectral condition does not imply polynomial reproduction.

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.