Sobolev Norm Convergence of Stationary Subdivision Schemes X1. Introduction

We show that Sobolev norm convergence of a stationary subdivision scheme is equivalent to standard norm convergence of the stationary subdivision scheme to a limit function in the Sobolev space. Stationary subdivision is an iterative method for the generation of reenable functions which are the main building blocks for the construction of wavelets. For this reason and also because of its use in geometric modeling stationary subdivision is a subject studied in its own right. Indeed, it is well known that whenever a stationary subdivision scheme converges in L p (IR s), it generates the solution of a reenement equation. Conversely, any stable solution of a reenement equation induces a convergent stationary subdivision scheme, see 2]. More detailed facts about univariate subdivision schemes, in particular about the regularity of the limit functions, can be found in 6]. In this paper we formulate the notion of convergence of stationary subdivision schemes in the Sobolev space W n p (IR s), and show that a stationary subdivision scheme converges in W n p (IR s) if and only if it converges in W p (IR s) to a limit function which belongs to W n p (IR s). All the results are proved in the more general context of vector subdivision. x2. Notions and Notation For n 2 IN we let ZZ n denote the set f0;1;:::;n ? 1g. We nd it convenient to use this set for indexing the components of vectors. Thus we will express x 2 IR s as x = (x j : j 2 ZZ s), where each x j 2 IR, j 2 ZZ s. In a similar way, m n matrices will be written as A = (A jk : j 2 ZZ m ; k 2 ZZ n). Also, let j j p denote the standard p{norm on IR s. For the unit vectors in IR s we will write