Hölder Regularity of Geometric Subdivision Schemes

Abstract

We present a framework for analyzing non-linear R d -valued subdivision schemes which are geometric in the sense that they commute with similarities in R d . It admits to establish C 1,α -regularity for arbitrary schemes of this type, and C 2,α -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.

Convergence of geometric subdivision schemes

“…Hence, commutation with rotations is not needed and the class of schemes for which the convergence analysis applies is even larger. But for consistency with [13] and because commutation with rotations is anyway a natural property for geometric subdivision, we stay with this definition. We also note that continuity in n m+1 is only demanded for g 0 .…”

“…Because the analysis presented here can be regarded as a generalization of [13], we adapt the setting and repeat the notation used there. Our analysis applies not only to the plane as in [12] but to arbitrary dimensions.…”

“…However, a general framework for manifold-valued subdivision is developed in [8][9][10][11] focusing on adaption of linear schemes to non-linear geometry. For non-linear schemes in a linear space, there are at least two contributions aiming at general results, namely [12] and [13].…”

“…This paper can be seen as a continuation of both, [12] and [13]: on one hand, we take the class of GLUE-schemes and skip equilinearity to obtain a broader set of schemes called GLU-schemes. In particular, dropping equilinearity is a substantial generalization as it allows to treat some uncovered schemes as GLU-schemes, like Example 2.3.…”

“…In the second one, [13], a broad class of subdivision schemes, called GLUE-schemes, is introduced. These schemes have to be geometric (i.e., commute with similarities), local (i.e., new points depend only on a fixed number of old points), uniform (i.e., the same rules are applied everywhere) and equilinear (i.e., linear polygons are mapped to linear polygons with half spacing), which is abbreviated in the acronym GLUE.…”