A generalized Taylor factorization for Hermite subdivision schemes

In a recent paper, we investigated factorization properties of Hermite subdivision schemes by means of the so–called Taylor factorization. This decomposition is based on a spectral condition which is satisfied for example by all interpolatory Hermite schemes. Nevertheless, there exist examples of Hermite schemes, especially some based on cardinal splines, which fail the spectral condition. For these schemes (and others) we provide the concept of an generalized Taylor factorization and show how it can be used to obtain convergence criteria for the Hermite scheme by means of factorization and contractivity.