Vector multivariate subdivision schemes: Comparison of spectral methods for their regularity analysis

We study vector multivariate subdivision schemes with dilation 2I satisfying sum rules of order k+1 and multiplicity m. It is well known that the magnitude of the associated joint spectral radius or, alternatively, the magnitude of the associated restricted spectral radius characterizes the -regularity, k∈N0, 1⩽p⩽∞, of such a scheme. This characterization alone does not necessarily indicate any intrinsic connection between the two radii. In this paper, we unify the two approaches based on the concepts of the joint spectral radius and the restricted spectral radius and show that these two numbers are equal. Therefore, the only difference between these approaches is that they offer different numerical schemes for estimating the regularity of subdivision. We show how to obtain the restricted spectral radius estimates using the techniques of linear programming and convex minimization. We illustrate our results with several examples.