  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.
