Analysis and Convergence of Hermite Subdivision Schemes

Abstract

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.