Increasing the smoothness of vector and Hermite subdivision schemes

TL;DR

This paper introduces a method to enhance the smoothness of vector and Hermite subdivision schemes by manipulating their symbols, enabling the construction of schemes with arbitrarily high regularity.

Contribution

It extends the scalar subdivision smoothing technique to vector and Hermite schemes through symbol manipulation, allowing higher regularity construction.

Findings

Method successfully increases scheme smoothness by one level.

Algorithms enable construction of schemes with arbitrarily high regularity.

Applicable to vector and Hermite subdivision schemes.

Abstract

In this paper we suggest a method for transforming a vector subdivision scheme generating $C^{\ell}$ limits to another such scheme of the same dimension, generating $C^{\ell+1}$ limits. In scalar subdivision, it is well known that a scheme generating $C^{\ell}$ limit curves can be transformed to a new scheme producing $C^{\ell+1}$ limit curves by multiplying the scheme's symbol with the smoothing factor $\tfrac{z+1}{2}$. 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).