Improving the convergence analysis of linear subdivision schemes

TL;DR

This paper addresses the convergence analysis of binary, univariate, linear subdivision schemes by tying convergence speed to a contractivity factor $\mu$. It adopts a Laurent-polynomial framework with symbols $a(z)$ and $q(z)$, using the relation $\Delta(S_a f)=S_q \Delta f$ and the condition $a(z)=(1+z) q(z)$ to study convergence through the operator $S_q$. A key theoretical result is that the contractivity factor of any convergent scheme cannot be smaller than $\tfrac{1}{2}$, with spline-generated schemes achieving this optimal rate via $q(z)=\left(\frac{1+z}{2}\right)^{m}$. The authors also propose an improved algorithm for determining convergence and discuss how to extend the approach to assess higher-order smoothness by examining scaled symbols such as $S_{2^n q}$ and the decomposition $a(z)=(1+z)^n q(z)$, enabling practical checks of $C^n$ regularity.

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.

Theorems & Definitions (15)

• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Corollary 3.3
• Remark 3.4
• Lemma 3.5
• Theorem 3.6
• proof
• Theorem 3.7