Increasing the smoothness of vector and Hermite subdivision schemes

In this paper we suggest a method for transforming a vector subdivision scheme generating $C^{\mathrm{\ell }}$ limits to another such scheme of the same dimension, generating $C^{\mathrm{\ell }+1}$ limits. In scalar subdivision, it is well known that a scheme generating $C^{\mathrm{\ell }}$ limit curves can be transformed to a new scheme producing $C^{\mathrm{\ell }+1}$ limit curves by multiplying the scheme’s symbol with the smoothing factor $\frac{z+1}{2}$. We extend this approach to vector and Hermite subdivision schemes, by manipulating symbols. The algorithms presented in this paper allow to construct vector (Hermite) subdivision schemes of arbitrarily high regularity from a convergent vector scheme (from a Hermite scheme whose Taylor scheme is convergent with limit functions of vanishing first component).