Multiple multivariate subdivision schemes: matrix and operator approaches

TL;DR

This paper generalizes the matrix approach to analyze the convergence of multiple multivariate subdivision schemes with level-dependent weights and dilation matrices, using joint spectral radius techniques.

Contribution

It extends the matrix-based convergence analysis to multiple subdivision schemes with level-dependent parameters, overcoming previous limitations.

Findings

Convergence characterized by joint spectral radius of derived matrices

Provides conditions for convergence of multiple subdivision schemes

Includes illustrative examples demonstrating the approach

Abstract

This paper extends the matrix based approach to the setting of multiple subdivision schemes studied in [Sauer 2012]. Multiple subdivision schemes, in contrast to stationary and non-stationary schemes, allow for level dependent subdivision weights and for level dependent choice of the dilation matrices. The latter property of multiple subdivision makes the standard definition of the transition matrices, crucial ingredient of the matrix approach in the stationary and non-stationary settings, inapplicable. We show how to avoid this obstacle and characterize the convergence of multiple subdivision schemes in terms of the joint spectral radius of certain square matrices derived from subdivision weights. We illustrate our results with several examples.