Generalized Taylor operators and polynomial chains for Hermite subdivision schemes

Abstract

Hermite subdivision schemes act on vector valued data that is not only considered as functions values of a vector valued function from $\mathbb {R}$ to $\mathbb {R}^r$, but as evaluations of r consecutive derivatives of a function. This intuition leads to a mild form of level dependence of the scheme. Previously, we have proved that a property called spectral condition or sum rule implies a factorization in terms of a generalized difference operator that gives rise to a “difference scheme” whose contractivity governs the convergence of the scheme. But many convergent Hermite schemes, for example, those based on cardinal splines, do not satisfy the spectral condition. In this paper, we generalize the property in a way that preserves all the above advantages: the associated factorizations and convergence theory. Based on these results, we can include the case of cardinal splines in a systematic way and are also able to construct new types of convergent Hermite subdivision schemes.