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.

翻译：本文针对二元、单变量线性细分格式的收敛性分析提出了若干新结果，这些结果均与收敛格式的{\it 压缩因子}相关。首先，我们证明收敛格式的压缩因子不可能低于二分之一。由于该因子越小，格式的收敛速度越快，因此具有二分之一压缩因子的格式（例如生成样条函数的格式）具有最优收敛速率。此外，我们进一步提出了线性格式收敛的深入见解与条件，并通过改进的算法展示了这些结论在判定此类细分格式收敛性中的实际应用。