Multivariate Generalized Hermite Subdivision Schemes

TL;DR

This paper introduces generalized Hermite subdivision schemes, characterizes their properties, and proves the existence of smooth, convergent schemes with polynomial-interpolation features, extending classical univariate results.

Contribution

It defines a new class of generalized Hermite subdivision schemes, analyzes their convergence and smoothness, and constructs schemes with desired properties, broadening the scope of existing subdivision theory.

Findings

Existence of convergent smooth schemes with linear-phase moments

Characterization of convergence and smoothness conditions

Extension of classical univariate Hermite subdivision results

Abstract

Due to properties such as interpolation, smoothness, and spline connections, Hermite subdivision schemes employ fast iterative algorithms for geometrically modeling curves/surfaces in CAGD and for building Hermite wavelets in numerical PDEs. In this paper we introduce a notion of generalized Hermite (dyadic) subdivision schemes and then we characterize their convergence, smoothness and underlying matrix masks with or without interpolation properties. We also introduce the notion of linear-phase moments for achieving the polynomial-interpolation property. For any given positive integer m, we constructively prove that there always exist convergent smooth generalized Hermite subdivision schemes with linear-phase moments such that their basis vector functions are spline functions in $C^m$ and have linearly independent integer shifts. As byproducts, our results resolve convergence, smoothness and existence of Lagrange, Hermite, or Birkhoff subdivision schemes. Even in dimension one our results significantly generalize and extend many known results on extensively studied univariate Hermite subdivision schemes. To illustrate the theoretical results in this paper, we provide examples of convergent generalized Hermite subdivision schemes with symmetric matrix masks having short support and smooth basis vector functions with or without interpolation property.