Multivariate Vector Subdivision Schemes with a General Matrix-valued Filter

TL;DR

This paper introduces a unified framework for analyzing the convergence of multivariate vector subdivision schemes with general matrix-valued filters, extending beyond classical Lagrange and Hermite schemes, with theoretical insights and practical examples.

Contribution

It defines a unique meaningful way to characterize vector subdivision schemes with general matrix filters and links their convergence to the vector cascade algorithm.

Findings

Unified convergence analysis for general matrix-valued filters

Extension of existing convergence results for classical schemes

Examples illustrating the theoretical framework

Abstract

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.