Generalized Taylor operators and Hermite subdivision schemes

TL;DR

This paper extends the theory of Hermite subdivision schemes by generalizing the spectral condition, allowing for the inclusion of schemes like cardinal splines and enabling the construction of new convergent schemes.

Contribution

The authors generalize the spectral condition for Hermite schemes, preserving factorization and convergence properties, and facilitating the development of new schemes including cardinal splines.

Findings

Generalized the spectral condition for Hermite schemes.

Established convergence theory for the generalized property.

Enabled construction of new convergent Hermite schemes.

Abstract

Hermite subdivision schemes act on vector valued data that is not only considered as functions values in $\mathbb{R}^r$, but as consecutive derivatives, which 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 and also enables us to construct new types of convergent Hermite subdivision schemes.