Non-stationary multivariate subdivision: joint spectral radius and asymptotic similarity

Abstract

In this paper we study scalar multivariate non-stationary subdivision schemes with a general integer dilation matrix. We present a new numerically efficient method for checking convergence and Hölder regularity of such schemes. This method relies on the concepts of approximate sum rules, asymptotic similarity and the so-called joint spectral radius of a finite set of square matrices. The combination of these concepts allows us to employ recent advances in linear algebra for exact computation of the joint spectral radius that have had already a great impact on studies of stationary subdivision schemes. We also expose the limitations of non-stationary schemes in their capability to reproduce and generate certain function spaces. We illustrate our results with several examples.