This work presents several new results concerning the analysis of the convergence of binary, univariate, and linear subdivision schemes, all related to the 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 the scheme's convergence, and schemes with contractivity factor $\frac{1}{2}$, such as those generating spline functions, have optimal convergence rates. 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.

翻译：本文提出了关于二元、单变量及线性细分格式收敛性分析的若干新结果，这些结果均与收敛格式的收缩因子相关。首先，我们证明了一个收敛格式的收缩因子不能低于二分之一。由于该因子越低，格式收敛速度越快，且具有收缩因子$\frac{1}{2}$的格式（例如生成样条函数的格式）具有最优收敛速率。此外，我们进一步揭示了线性格式收敛的深层条件与性质，并展示了一种改进算法在判定此类细分格式收敛性中的适用性。