Hölder Regularity of Geometric Subdivision Schemes

Abstract

We present a framework for analyzing non-linear ${\mathbb{R}}^d$-valued subdivision schemes which are geometric in the sense that they commute with similarities in ${\mathbb{R}}^d$. It admits to establish $C^{1,\alpha }$-regularity for arbitrary schemes of this type, and $C^{2,\alpha }$-regularity for an important subset thereof, which includes all real-valued schemes. Our results are constructive in the sense that they can be verified explicitly for any scheme and any given set of initial data by a universal procedure. This procedure can be executed automatically and rigorously by a computer when using interval arithmetics.

31 pages, 1 figure