Regularity of Non-Stationary Multivariate Subdivision

In this paper, we study scalar multivariate non-stationary subdivision schemes with integer dilation matrixM = mI, m ≥ 2, and present a general approach for checking their convergence and for determining their Hölder regularity. The combination of the concepts of asymptotic similarity and approximate sum rules allows us to link stationary and nonstationary settings and to employ recent advances in methods for exact computation of the joint spectral radius. As an application, we prove a recent conjecture on the Hölder regularity of the generalized Daubechies wavelets. We illustrate our results with several examples. We also expose limitations of non-stationary schemes in their capability to reproduce and generate certain function spaces.