Improving the convergence analysis of linear subdivision schemes

TL;DR

This paper improves the understanding of convergence in linear subdivision schemes by establishing a lower bound on the contractivity factor and providing new conditions for convergence analysis.

Contribution

It introduces a lower bound of 1/2 for the contractivity factor in convergent schemes and offers enhanced methods for convergence verification.

Findings

Convergent schemes cannot have a contractivity factor less than 1/2.

Schemes with a contractivity factor of 1/2, like spline-generating schemes, have optimal convergence rates.

New conditions and algorithms improve convergence analysis of linear subdivision schemes.

Abstract

This work presents several new results concerning the analysis of the convergence of binary, univariate, and linear subdivision schemes, all related to the {\it contractivity factor} of a convergent scheme. First, we prove that a convergent scheme cannot have a contractivity factor lower than half. Since the lower this factor is, the faster is the convergence of the scheme, schemes with contractivity factor $\frac{1}{2}$, such as those generating spline functions, have optimal convergence rate. Additionally, we provide further insights and conditions for the convergence of linear schemes and demonstrate their applicability in an improved algorithm for determining the convergence of such subdivision schemes.