Subdivision schemes are closely related to splines and wavelets and have numerous applications in CAGD and numerical differential equations. Subdivision schemes employ a scalar filter; that is, scalar subdivision schemes, have been extensively studied in the literature. In contrast, subdivision schemes with a matrix filter, which are the so-called vector subdivision schemes, are far from being well understood. So far, only vector subdivision schemes that use special matrix-valued filters have been well-investigated, such as the Lagrange and Hermite subdivision schemes. To the best of our knowledge, it remains unclear how to define and characterize the convergence of a vector subdivision scheme that uses a general matrix-valued filter. Though filters from Lagrange and Hermite subdivision schemes have nice properties and are widely used in practice, filters not from either subdivision scheme appear in many applications. Hence, it is necessary to study vector subdivision schemes with a general matrix-valued filter. In this paper, from the perspective of a vector cascade algorithm, we show that there is only one meaningful way to define a vector subdivision scheme. We will analyze the convergence of the newly defined vector subdivision scheme and show that it is equivalent to the convergence of the corresponding vector cascade algorithm. Applying our theory, we show that existing results on the convergence of Lagrange and Hermite subdivision schemes can be easily obtained and improved. Finally, we will present some examples of vector subdivision schemes to illustrate our main results.

翻译：细分格式与样条及小波理论密切相关，在计算机辅助几何设计与数值微分方程领域具有广泛应用。采用标量滤波器的细分格式（即标量细分格式）已在文献中得到广泛研究。相比之下，采用矩阵滤波器的细分格式（即向量细分格式）尚未获得充分理解。迄今为止，仅使用特殊矩阵值滤波器的向量细分格式（如拉格朗日与埃尔米特细分格式）得到了深入研究。据我们所知，如何定义和刻画采用一般矩阵值滤波器的向量细分格式的收敛性仍不明确。尽管拉格朗日与埃尔米特细分格式的滤波器具有良好的性质并在实践中广泛应用，但非源于这两类细分格式的滤波器同样出现在众多应用场景中。因此，研究具有一般矩阵值滤波器的向量细分格式具有必要性。本文从向量级联算法的视角出发，论证了定义向量细分格式仅存在一种具有数学意义的方式。我们将分析新定义的向量细分格式的收敛性，并证明其等价于对应向量级联算法的收敛性。应用本文理论，我们展示了现有关于拉格朗日与埃尔米特细分格式收敛性的结果可被简便地推导与改进。最后，我们将通过若干向量细分格式的算例来阐释主要结论。