Hermite interpolation property is desired in applied and computational mathematics. Hermite and vector subdivision schemes are of interest in CAGD for generating subdivision curves and in computational mathematics for building Hermite wavelets to numerically solve partial differential equations. In contrast to well-studied scalar subdivision schemes, Hermite and vector subdivision schemes employ matrix-valued masks and vector input data, which make their analysis much more complicated and difficult than their scalar counterparts. Under the spectral condition or the spectral chain, analysis of Hermite subdivision schemes through factorization of matrix-valued masks has been extensively studied in the literature and sufficient conditions have been given for the convergence of Hermite subdivision schemes through the contractivity of their derived subdivision schemes. We contribute to the study of Hermite subdivision schemes from a different perspective by investigating vector subdivision operators acting on vector polynomials and by establishing connections among Hermite subdivision schemes, vector cascade algorithms, and refinable vector functions. This approach allows us to characterize and construct all masks for Hermite subdivision schemes, to explain the spectral condition and spectral chain in the literature, to characterize convergence and smoothness of Hermite subdivision schemes using vector cascade algorithms, and to provide simple factorizations of Hermite masks through the normal form of matrix-valued masks such that the Hermite subdivision scheme is convergent if and only if its derived subdivision scheme is contractive. We also constructively prove that there always exist arbitrarily smooth convergent Hermite subdivision schemes, whose basis vector functions are splines and have linearly independent shifts. Several examples of Hermite subdivision schemes with short support and high smoothness are presented to illustrate the results in this paper.

Hermite 插值特性在应用和计算数学中是需要的。Hermite 和矢量细分方案在 CAGD 中用于生成细分曲线，在计算数学中用于构建 Hermite 小波以数值求解偏微分方程。与经过充分研究的标量细分方案相比，Hermite 和向量细分方案使用矩阵值掩码和矢量输入数据，这使得它们的分析比标量对应物更加复杂和困难。在光谱条件或光谱链下，通过矩阵值掩码的因式分解对 Hermite 细分方案的分析在文献中得到了广泛的研究，并且已经给出了通过其派生细分方案的收缩性来收敛 Hermite 细分方案的充分条件。我们通过研究作用于向量多项式的向量细分算子，并通过在 Hermite 细分方案、向量级联算法和可优化向量函数之间建立联系，从不同的角度对 Hermite 细分方案的研究做出贡献。这种方法允许我们表征和构造 Hermite 细分方案的所有掩码，解释文献中的光谱条件和光谱链，使用向量级联算法表征 Hermite 细分方案的收敛性和平滑性，并通过矩阵值掩码的标准形式提供 Hermite 掩码的简单分解，使得 Hermite 细分方案收敛当且仅当其派生的细分方案是收缩的。我们还建设性地证明了总是存在任意平滑收敛 Hermite 细分方案，其基向量函数是样条并具有线性无关的位移。给出了几个具有短支撑和高平滑度的 Hermite 细分方案的例子来说明本文的结果。我们还建设性地证明了总是存在任意平滑收敛 Hermite 细分方案，其基向量函数是样条并具有线性无关的位移。给出了几个具有短支撑和高平滑度的 Hermite 细分方案的例子来说明本文的结果。我们还建设性地证明了总是存在任意平滑收敛 Hermite 细分方案，其基向量函数是样条并具有线性无关的位移。给出了几个具有短支撑和高平滑度的 Hermite 细分方案的例子来说明本文的结果。